A108299 Triangle read by rows, 0 <= k <= n: T(n,k) = binomial(n-[(k+1)/2],[k/2])*(-1)^[(k+1)/2].

1, 1, -1, 1, -1, -1, 1, -1, -2, 1, 1, -1, -3, 2, 1, 1, -1, -4, 3, 3, -1, 1, -1, -5, 4, 6, -3, -1, 1, -1, -6, 5, 10, -6, -4, 1, 1, -1, -7, 6, 15, -10, -10, 4, 1, 1, -1, -8, 7, 21, -15, -20, 10, 5, -1, 1, -1, -9, 8, 28, -21, -35, 20, 15, -5, -1, 1, -1, -10, 9, 36, -28, -56, 35, 35, -15, -6, 1, 1, -1, -11, 10, 45, -36, -84, 56, 70

Offset

0,9

Comments

Matrix inverse of A124645.

Let L(n,x) = Sum_{k=0..n} T(n,k)*x^(n-k) and Pi=3.14...:

L(n,x) = Product_{k=1..n} (x - 2*cos((2*k-1)*Pi/(2*n+1)));

Sum_{k=0..n} T(n,k) = L(n,1) = A010892(n+1);

Sum_{k=0..n} abs(T(n,k)) = A000045(n+2);

abs(T(n,k)) = A065941(n,k), T(n,k) = A065941(n,k)*A087960(k);

T(2*n,k) + T(2*n+1,k+1) = 0 for 0 <= k <= 2*n;

T(n,0) = A000012(n) = 1; T(n,1) = -1 for n > 0;

T(n,2) = -(n-1) for n > 1; T(n,3) = A000027(n)=n for n > 2;

T(n,4) = A000217(n-3) for n > 3; T(n,5) = -A000217(n-4) for n > 4;

T(n,6) = -A000292(n-5) for n > 5; T(n,7) = A000292(n-6) for n > 6;

T(n,n-3) = A058187(n-3)*(-1)^floor(n/2) for n > 2;

T(n,n-2) = A008805(n-2)*(-1)^floor((n+1)/2) for n > 1;

T(n,n-1) = A008619(n-1)*(-1)^floor(n/2) for n > 0;

T(n,n) = L(n,0) = (-1)^floor((n+1)/2);

L(n,1) = A010892(n+1); L(n,-1) = A061347(n+2);

L(n,2) = 1; L(n,-2) = A005408(n)*(-1)^n;

L(n,3) = A001519(n); L(n,-3) = A002878(n)*(-1)^n;

L(n,4) = A001835(n+1); L(n,-4) = A001834(n)*(-1)^n;

L(n,5) = A004253(n); L(n,-5) = A030221(n)*(-1)^n;

L(n,6) = A001653(n); L(n,-6) = A002315(n)*(-1)^n;

L(n,7) = A049685(n); L(n,-7) = A033890(n)*(-1)^n;

L(n,8) = A070997(n); L(n,-8) = A057080(n)*(-1)^n;

L(n,9) = A070998(n); L(n,-9) = A057081(n)*(-1)^n;

L(n,10) = A072256(n+1); L(n,-10) = A054320(n)*(-1)^n;

L(n,11) = A078922(n+1); L(n,-11) = A097783(n)*(-1)^n;

L(n,12) = A077417(n); L(n,-12) = A077416(n)*(-1)^n;

L(n,13) = A085260(n);

L(n,14) = A001570(n); L(n,-14) = A028230(n)*(-1)^n;

L(n,n) = A108366(n); L(n,-n) = A108367(n).

Row n of the matrix inverse (A124645) has g.f.: x^floor(n/2)*(1-x)^(n-floor(n/2)). - Paul D. Hanna, Jun 12 2005

From L. Edson Jeffery, Mar 12 2011: (Start)

Conjecture: Let N=2*n+1, with n > 2. Then T(n,k) (0 <= k <= n) gives the k-th coefficient in the characteristic function p_N(x)=0, of degree n in x, for the n X n tridiagonal unit-primitive matrix G_N (see [Jeffery]) of the form

G_N=A_{N,1}=

(0 1 0 ... 0)

(1 0 1 0 ... 0)

(0 1 0 1 0 ... 0)

...

(0 ... 0 1 0 1)

(0 ... 0 1 1),

with solutions phi_j = 2*cos((2*j-1)*Pi/N), j=1,2,...,n. For example, for n=3,

G_7=A_{7,1}=

(0 1 0)

(1 0 1)

(0 1 1).

We have {T(3,k)}=(1,-1,-2,1), while the characteristic function of G_7 is p(x) = x^3-x^2-2*x+1 = 0, with solutions phi_j = 2*cos((2*j-1)*Pi/7), j=1,2,3. (End)

The triangle sums, see A180662 for their definitions, link A108299 with several sequences, see the crossrefs. - Johannes W. Meijer, Aug 08 2011

The roots to the polynomials are chaotic using iterates of the operation (x^2 - 2), with cycle lengths L and initial seeds returning to the same term or (-1)* the seed. Periodic cycle lengths L are shown in A003558 such that for the polynomial represented by row r, the cycle length L is A003558(r-1). The matrices corresponding to the rows as characteristic polynomials are likewise chaotic [cf. Kappraff et al., 2005] with the same cycle lengths but substituting 2*I for the "2" in (x^2 - 2), where I = the Identity matrix. For example, the roots to x^3 - x^2 - 2x + 1 = 0 are 1.801937..., -1.246979..., and 0.445041... With 1.801937... as the initial seed and using (x^2 - 2), we obtain the 3-period trajectory of 8.801937... -> 1.246979... -> -0.445041... (returning to -1.801937...). We note that A003558(2) = 3. The corresponding matrix M is: [0,1,0; 1,0,1; 0,1,1,]. Using seed M with (x^2 - 2*I), we obtain the 3-period with the cycle completed at (-1)*M. - Gary W. Adamson, Feb 07 2012

References

Friedrich L. Bauer, 'De Moivre und Lagrange: Cosinus eines rationalen Vielfachen von Pi', Informatik Spektrum 28 (Springer, 2005).

Jay Kappraff, S. Jablan, G. Adamson, & R. Sazdonovich: "Golden Fields, Generalized Fibonacci Sequences, & Chaotic Matrices"; FORMA, Vol 19, No 4, (2005).

Links

Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened

Henry W. Gould, A Variant of Pascal's Triangle, Corrections, The Fibonacci Quarterly, Vol. 3, Nr. 4, Dec. 1965, p. 257-271.

L. Edson Jeffery, Unit-primitive matrices.

Ju, Hyeong-Kwan On the sequence generated by a certain type of matrices. Honam Math. J. 39, No. 4, 665-675 (2017).

Michelle Rudolph-Lilith, On the Product Representation of Number Sequences, with Application to the Fibonacci Family, arXiv preprint arXiv:1508.07894 [math.NT], 2015.

Frank Ruskey and Carla Savage, Gray codes for set partitions and restricted growth tails, Australasian Journal of Combinatorics, Volume 10(1994), pp. 85-96. See Table 1 p. 95.

Formula

T(n,k) = binomial(n-floor((k+1)/2),floor(k/2))*(-1)^floor((k+1)/2).

T(n+1, k) = if sign(T(n, k-1))=sign(T(n, k)) then T(n, k-1)+T(n, k) else -T(n, k-1) for 0 < k < n, T(n, 0) = 1, T(n, n) = (-1)^floor((n+1)/2).

G.f.: A(x, y) = (1 - x*y)/(1 - x + x^2*y^2). - Paul D. Hanna, Jun 12 2005

The generating polynomial (in z) of row n >= 0 is (u^(2*n+1) + v^(2*n+1))/(u + v), where u and v are defined by u^2 + v^2 = 1 and u*v = z. - Emeric Deutsch, Jun 16 2011

From Johannes W. Meijer, Aug 08 2011: (Start)

abs(T(n,k)) = A065941(n,k) = abs(A187660(n,n-k));

T(n,n-k) = A130777(n,k); abs(T(n,n-k)) = A046854(n,k) = abs(A066170(n,k)). (End)

Example

Triangle begins:
  1;
  1,  -1;
  1,  -1,  -1;
  1,  -1,  -2,   1;
  1,  -1,  -3,   2,   1;
  1,  -1,  -4,   3,   3,  -1;
  1,  -1,  -5,   4,   6,  -3,  -1;
  1,  -1,  -6,   5,  10,  -6,  -4,   1;
  1,  -1,  -7,   6,  15, -10, -10,   4,   1;
  1,  -1,  -8,   7,  21, -15, -20,  10,   5,  -1;
  1,  -1,  -9,   8,  28, -21, -35,  20,  15,  -5,  -1;
  1,  -1, -10,   9,  36, -28, -56,  35,  35, -15,  -6,   1;
  ...

Maple

A108299 := proc(n, k): binomial(n-floor((k+1)/2), floor(k/2))*(-1)^floor((k+1)/2) end: seq(seq(A108299 (n, k), k=0..n), n=0..11); # Johannes W. Meijer, Aug 08 2011

Mathematica

t[n_, k_?EvenQ] := I^k*Binomial[n-k/2, k/2]; t[n_, k_?OddQ] := -I^(k-1)*Binomial[n+(1-k)/2-1, (k-1)/2]; Table[t[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 16 2013 *)

Prog

(PARI) {T(n, k)=polcoeff(polcoeff((1-x*y)/(1-x+x^2*y^2+x^2*O(x^n)), n, x)+y*O(y^k), k, y)} (Hanna)
(Haskell)
a108299 n k = a108299_tabl !! n !! k
a108299_row n = a108299_tabl !! n
a108299_tabl = [1] : iterate (\row ->
   zipWith (+) (zipWith (*) ([0] ++ row) a033999_list)
               (zipWith (*) (row ++ [0]) a059841_list)) [1, -1]
-- Reinhard Zumkeller, May 06 2012

Crossrefs

Cf. A049310, A039961, A124645 (matrix inverse).

Triangle sums (see the comments): A193884 (Kn11), A154955 (Kn21), A087960 (Kn22), A000007 (Kn3), A010892 (Fi1), A134668 (Fi2), A078031 (Ca2), A193669 (Gi1), A001519 (Gi3), A193885 (Ze1), A050935 (Ze3). - Johannes W. Meijer, Aug 08 2011

Cf. A003558.

Cf. A033999, A059841.

Keyword

sign,tabl

Author

Reinhard Zumkeller, Jun 01 2005

Extensions

Corrected and edited by Philippe Deléham, Oct 20 2008

Status

approved

A112468 Riordan array (1/(1-x), x/(1+x)).

1, 1, 1, 1, 0, 1, 1, 1, -1, 1, 1, 0, 2, -2, 1, 1, 1, -2, 4, -3, 1, 1, 0, 3, -6, 7, -4, 1, 1, 1, -3, 9, -13, 11, -5, 1, 1, 0, 4, -12, 22, -24, 16, -6, 1, 1, 1, -4, 16, -34, 46, -40, 22, -7, 1, 1, 0, 5, -20, 50, -80, 86, -62, 29, -8, 1, 1, 1, -5, 25, -70, 130, -166, 148, -91, 37, -9, 1, 1, 0, 6, -30, 95, -200, 296, -314, 239, -128, 46, -10, 1

Offset

0,13

Comments

Row sums are A040000. Diagonal sums are A112469. Inverse is A112467. Row sums of k-th power are 1, k+1, k+1, k+1, .... Note that C(n,k) = Sum_{j=0..n-k} C(n-j-1, n-k-j).

Equals row reversal of triangle A112555 up to sign, where log(A112555) = A112555 - I. Unsigned row sums equals A052953 (Jacobsthal numbers + 1). Central terms of even-indexed rows are a signed version of A072547. Sums of squared terms in rows yields A112556, which equals the first differences of the unsigned central terms. - Paul D. Hanna, Jan 20 2006

Sum_{k=0..n} T(n,k)*x^k = A000012(n), A040000(n), A005408(n), A033484(n), A048473(n), A020989(n), A057651(n), A061801(n), A238275(n), A238276(n), A138894(n), A090843(n), A199023(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 respectively (see the square array in A112739). - Philippe Deléham, Feb 22 2014

Links

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

H. Belbachir and F. Bencherif, On some properties of bivariate Fibonacci and Lucas Polynomials, JIS 11 (2008) 08.2.6.

Hacene Belbachir and Athmane Benmezai, Expansion of Fibonacci and Lucas Polynomials: An Answer to Prodinger's Question, Journal of Integer Sequences, Vol. 15 (2012), #12.7.6.

Emeric Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Applied Mathematics, 34 (2005) pp. 101-122.

Kyu-Hwan Lee and Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.

H. Prodinger, On the expansion of Fibonacci and Lucas Polynomials, JIS 12 (2009) 09.1.6.

Formula

Triangle T(n,k) read by rows: T(n,0)=1, T(n,k) = T(n-1,k-1) - T(n-1,k). - Mats Granvik, Mar 15 2010

Number triangle T(n, k)= Sum_{j=0..n-k} C(n-j-1, n-k-j)*(-1)^(n-k-j).

G.f. of matrix power T^m: (1+(m-1)*x)*(1+m*x)/(1+m*x-x*y)/(1-x). G.f. of matrix log: x*(1-2*x*y+x^2*y)/(1-x*y)^2/(1-x). - Paul D. Hanna, Jan 20 2006

T(n, k) = R(n,n-k,-1) where R(n,k,m) = (1-m)^(-n+k)-m^(k+1)*Pochhammer(n-k,k+1)*hyper2F1([1,n+1],[k+2],m)/(k+1)!. - Peter Luschny, Jul 25 2014

Example

Triangle starts
  1;
  1,  1;
  1,  0,  1;
  1,  1, -1,  1;
  1,  0,  2, -2,  1;
  1,  1, -2,  4, -3,  1;
  1,  0,  3, -6,  7, -4,  1;
Matrix log begins:
  0;
  1,  0;
  1,  0,  0;
  1,  1, -1,  0;
  1,  1,  1, -2,  0;
  1,  1,  1,  1, -3,  0; ...
Production matrix begins
  1,  1,
  0, -1,  1,
  0,  0, -1,  1,
  0,  0,  0, -1,  1,
  0,  0,  0,  0, -1,  1,
  0,  0,  0,  0,  0, -1,  1,
  0,  0,  0,  0,  0,  0, -1,  1.
- Paul Barry, Apr 08 2011

Maple

T := (n, k, m) -> (1-m)^(-n+k)-m^(k+1)*pochhammer(n-k, k+1)*hypergeom( [1, n+1], [k+2], m)/(k+1)!; A112468 := (n, k) -> T(n, n-k, -1);
seq(print(seq(simplify(A112468(n, k)), k=0..n)), n=0..10); # Peter Luschny, Jul 25 2014

Mathematica

T[n_, 0] = 1; T[n_, n_] = 1; T[n_, k_ ]:= T[n, k] = T[n-1, k-1] - T[n-1, k]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* Jean-François Alcover, Mar 06 2013 *)

Prog

(PARI) {T(n, k)=local(m=1, x=X+X*O(X^n), y=Y+Y*O(Y^k)); polcoeff(polcoeff((1+(m-1)*x)*(1+m*x)/(1+m*x-x*y)/(1-x), n, X), k, Y)} \\ Paul D. Hanna, Jan 20 2006
(Haskell)
a112468 n k = a112468_tabl !! n !! k
a112468_row n = a112468_tabl !! n
a112468_tabl = iterate (\xs -> zipWith (-) ([2] ++ xs) (xs ++ [0])) [1]
-- Reinhard Zumkeller, Jan 03 2014
(PARI) T(n, k) = if(k==0 || k==n, 1, T(n-1, k-1) - T(n-1, k)); \\ G. C. Greubel, Nov 13 2019
(Magma)
function T(n, k)
  if k eq 0 or k eq n then return 1;
  else return T(n-1, k-1) - T(n-1, k);
  end if;
  return T;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 13 2019
(Sage)@CachedFunction
def T(n, k):
    if (k<0 or n<0): return 0
    elif (k==0 or k==n): return 1
    else: return T(n-1, k-1) - T(n-1, k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 13 2019
(GAP)
T:= function(n, k)
    if k=0 or k=n then return 1;
    else return T(n-1, k-1) - T(n-1, k);
    fi;
  end;
Flat(List([0..12], n-> List([0..n], k-> T(n, k) ))); # G. C. Greubel, Nov 13 2019

Crossrefs

Cf. A174294, A174295, A174296, A174297. - Mats Granvik, Mar 15 2010

Cf. A072547 (central terms), A112555 (reversed rows), A112465, A052953, A112556, A112739, A119258.

See A279006 for another version.

Keyword

easy,sign,tabl

Author

Paul Barry, Sep 06 2005

Status

approved

A055870 Signed Fibonomial triangle.

1, 1, -1, 1, -1, -1, 1, -2, -2, 1, 1, -3, -6, 3, 1, 1, -5, -15, 15, 5, -1, 1, -8, -40, 60, 40, -8, -1, 1, -13, -104, 260, 260, -104, -13, 1, 1, -21, -273, 1092, 1820, -1092, -273, 21, 1, 1, -34, -714, 4641, 12376, -12376, -4641, 714, 34, -1, 1, -55, -1870, 19635, 85085, -136136, -85085, 19635, 1870, -55, -1

Offset

0,8

Comments

Row n+1 (n >= 1) of the signed triangle lists the coefficients of the recursion relation for the n-th power of Fibonacci numbers A000045: sum(a(n+1,m)*(F(k-m))^n,m=0..n+1) = 0, k >= n+1; inputs: (F(k))^n, k=0..n.

The inverse of the row polynomial p(n,x) := sum(a(n,m)*x^m,m=0..n) is the g.f. for the column m=n-1 of the Fibonomial triangle A010048.

The row polynomials p(n,x) factorize according to p(n,x)=G(n-1)*p(n-2,-x), with inputs p(0,x)= 1, p(1,x)= 1-x and G(n) := 1-L(n)*x+(-1)^n*x^2, with L(n)=A000032(n) (Lucas). (Derived from Riordan's result and Knuth's exercise).

The row polynomials are the characteristic polynomials of product of the binomial matrix binomial(i,j) and the exchange matrix J_n (matrix with 1's on the antidiagonal, 0 elsewhere). - Paul Barry, Oct 05 2004

References

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 1, pp. 84-5 and 492.

Links

Table of n, a(n) for n=0..65.

Katharine A. Ahrens, Combinatorial Applications of the k-Fibonacci Numbers: A Cryptographically Motivated Analysis, Ph. D. thesis, North Carolina State University (2020).

A. T. Benjamin, S. S. Plott, A combinatorial approach to fibonomial coefficients, Fib. Quart. 46/47 (1) (2008/9) 7-9.

A. Brousseau, A sequence of power formulas, Fib. Quart., 6 (1968), 81-83.

H. W. Gould, Extensions of the Hermite g.c.d. theorems for binomial coefficients, Fib Quart. 33 (1995) 386.

E. Kilic, The generalized Fibonomial matrix, Eur. J. Combinat. 31 (1) (2010) 193-209.

Ron Knott, The Fibonomials

Ewa Krot, An introduction to finite fibonomial calculus, Centr. Eur. J. Math. 2 (5) (2004) 754.

A. K. Kwasniewski, Fibonomial cumulative connection constants, arXiv:math/0406006 [math.CO], 2004-2009.

Phakhinkon Phunphayap, Various Problems Concerning Factorials, Binomial Coefficients, Fibonomial Coefficients, and Palindromes, Ph. D. Thesis, Silpakorn University (Thailand 2021).

Phakhinkon Phunphayap, Prapanpong Pongsriiam, Explicit Formulas for the p-adic Valuations of Fibonomial Coefficients, J. Int. Seq. 21 (2018), #18.3.1.

J. Riordan, Generating functions for powers of Fibonacci numbers, Duke. Math. J. 29 (1962) 5-12.

J. Seibert, P. Trojovsky, On some identities for the Fibonomial coefficients, Math. Slov. 55 (2005) 9-19.

P. Trojovsky, On some identities for the Fibonomial coefficients..., Discr. Appl. Math. 155 (15) (2007) 2017

Formula

a(n, m)=(-1)^floor((m+1)/2)*A010048(n, m). A010048(n, m)=: fibonomial(n, m).

G.f. for column m: (-1)^floor((m+1)/2)*x^m/p(m+1, x) with the row polynomial of the (signed) triangle: p(n, x) := sum(a(n, m)*x^m, m=0..n).

Example

Row polynomial for n=4: p(4,x)=1-3*x-6*x^2+3*x^3+x^4= (1+x-x^2)*(1-4*x-x^2). 1/p(4,x) is G.f. for A010048(n+3,3), n >= 0: {1,3,15,60,...}= A001655(n).
n=3: 1*(F(k))^3 - 3*(F(k-1))^3 - 6*(F(k-2))^3 + 3*(F(k-3))^3 + 1*(F(k-4))^3 = 0, k >= 4; inputs: (F(k))^3, k=0..3.
The triangle begins:
n\m 0   1     2    3     4      5     6    7   8   9
0   1
1   1  -1
2   1  -1    -1
3   1  -2    -2    1
4   1  -3    -6    3     1
5   1  -5   -15   15     5     -1
6   1  -8   -40   60    40     -8    -1
7   1 -13  -104  260   260   -104   -13    1
8   1 -21  -273 1092  1820  -1092  -273   21   1
9   1 -34  -714 4641 12376 -12376 -4641  714  34  -1
... [Wolfdieter Lang, Aug 06 2012; a(7,1) corrected, Oct 10 2012]

Maple

A055870 := proc(n, k)
    (-1)^floor((k+1)/2)*A010048(n, k) ;
end proc: # R. J. Mathar, Jun 14 2015

Mathematica

a[n_, m_] := {1, -1, -1, 1}[[Mod[m, 4] + 1]] * Product[ Fibonacci[n - j + 1] / Fibonacci[j], {j, 1, m}]; Table[a[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jul 05 2013 *)

Crossrefs

Cf. A010048, A000032, A000045, A001654-8, A056565-7. Row sums (signed): A055871, (unsigned) A056569.

Cf. A051159.

Central column: A003268.

Keyword

easy,sign,tabl

Author

Wolfdieter Lang, Jul 10 2000

Status

approved

A372441 Number of binary indices (binary weight) of n minus number of prime indices (bigomega) of n.

1, 0, 1, -1, 1, 0, 2, -2, 0, 0, 2, -1, 2, 1, 2, -3, 1, -1, 2, -1, 1, 1, 3, -2, 1, 1, 1, 0, 3, 1, 4, -4, 0, 0, 1, -2, 2, 1, 2, -2, 2, 0, 3, 0, 1, 2, 4, -3, 1, 0, 2, 0, 3, 0, 3, -1, 2, 2, 4, 0, 4, 3, 3, -5, 0, -1, 2, -1, 1, 0, 3, -3, 2, 1, 1, 0, 2, 1, 4, -3, -1

Offset

1,7

Comments

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

a(n) = A000120(n) - A001222(n).

Maple

f:= proc(n) convert(convert(n, base, 2), `+`)-numtheory:-bigomega(n) end proc:
map(f, [$1..100]); # Robert Israel, May 22 2024

Mathematica

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Length[bix[n]]-Length[prix[n]], {n, 100}]

Crossrefs

Positions of zeros are A071814.

For sum instead of length we have A372428, zeros A372427.

For minimum instead of length we have A372437, zeros {}.

For maximum instead of length we have A372442, zeros A372436.

Positions of odd terms are A372590, even A372591.

A003963 gives product of prime indices.

A019565 gives Heinz number of binary indices, adjoint A048675.

A029837 gives greatest binary index, least A001511.

A048793 lists binary indices, length A000120, reverse A272020, sum A029931.

A061395 gives greatest prime index, least A055396.

A070939 gives length of binary expansion.

A112798 lists prime indices, length A001222, reverse A296150, sum A056239.

Cf. A000720, A014499, A038802, A061712, A174090, A243055, A304818, A359495, A372429-A372432.

Keyword

sign,base

Author

Gus Wiseman, May 07 2024

Status

approved

A144431 Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n,1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = -1.

1, 1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -2, 2, -2, 1, 1, -3, 2, 2, -3, 1, 1, -4, 7, -8, 7, -4, 1, 1, -5, 9, -5, -5, 9, -5, 1, 1, -6, 16, -26, 30, -26, 16, -6, 1, 1, -7, 20, -28, 14, 14, -28, 20, -7, 1, 1, -8, 29, -64, 98, -112, 98, -64, 29, -8, 1, 1, -9, 35, -75, 90, -42, -42, 90, -75, 35, -9, 1

Offset

1,12

Comments

Row sums are: {1, 2, 2, 0, 0, 0, 0, 0, 0, 0, ...}.

For m = ...,-1,0,1,2 we get ..., A144431, A007318 (Pascal), A008292, A060187, ..., so this might be called a sub-Pascal triangle.

The triangle starts off like A098593, but is different further on.

Links

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 8.

Formula

T(n,k) = (m*n - m*k + 1)*T(n-1, k-1) + (m*k - (m-1))*T(n-1, k) with T(n, 1) = T(n, n) = 1 and m = -1.

From G. C. Greubel, Mar 01 2022: (Start)

T(n, n-k) = T(n, k).

T(n, k) = (-1)^(k-1)*binomial(n-3, k-1) + (-1)^(n+k)*binomial(n-3, k-3) with T(1, k) = T(2, k) = 1.

Sum_{k=1..n} T(n, k) = [n==1] + 2*[n==2] + 2*[n==3] + (1-(-1)^n)*0^(n-3)*[n>3]. (End)

Example

Triangle begins:
  1;
  1,   1;
  1,   0,   1;
  1,  -1,  -1,   1;
  1,  -2,   2,  -2,   1;
  1,  -3,   2,   2,  -3,   1;
  1,  -4,   7,  -8,   7,  -4,   1;
  1,  -5,   9,  -5,  -5,   9,  -5,   1;
  1,  -6,  16, -26,  30, -26,  16,  -6,   1;
  1,  -7,  20, -28,  14,  14, -28,  20,  -7,   1;
  ...

Maple

T:=proc(n, k, l) option remember;
if (n=1 or k=1 or k=n) then 1 else
(l*n-l*k+1)*T(n-1, k-1, l)+(l*k-l+1)*T(n-1, k, l); fi; end;
for n from 1 to 15 do lprint([seq(T(n, k, -1), k=1..n)]); od; # N. J. A. Sloane, May 08 2013

Mathematica

m=-1;
T[n_, 1]:= 1; T[n_, n_]:= 1;
T[n_, k_]:= (m*n-m*k+1)*T[n-1, k-1] + (m*k - (m - 1))*T[n-1, k];
Table[T[n, k], {n, 15}, {k, n}]//Flatten

Prog

(Sage)
def A144431(n, k):
    if (n<3): return 1
    else: return (-1)^(k-1)*binomial(n-3, k-1) + (-1)^(n+k)*binomial(n-3, k-3)
flatten([[A144431(n, k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 01 2022

Crossrefs

Cf. A007318, A008292, A060187, A098593.

Keyword

tabl,easy,sign

Author

Roger L. Bagula, Oct 04 2008

Extensions

Edited by N. J. A. Sloane, May 08 2013

Status

approved

A350942 Number of odd parts minus number of even conjugate parts of the integer partition with Heinz number n.

0, 1, 0, 1, 1, 0, 0, 3, -2, 1, 1, 2, 0, 0, -1, 3, 1, 0, 0, 3, -2, 1, 1, 2, -1, 0, 0, 2, 0, 1, 1, 5, -1, 1, -2, 0, 0, 0, -2, 3, 1, 0, 0, 3, 1, 1, 1, 4, -4, 1, -1, 2, 0, 0, -1, 2, -2, 0, 1, 1, 0, 1, 0, 5, -2, 1, 1, 3, -1, 0, 0, 2, 1, 0, 1, 2, -3, 0, 0, 5, -2, 1

Offset

1,8

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

Links

Table of n, a(n) for n=1..82.

Example

First positions n such that a(n) = 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, together with their prime indices, are:
  192: (2,1,1,1,1,1,1)
   32: (1,1,1,1,1)
   48: (2,1,1,1,1)
    8: (1,1,1)
   12: (2,1,1)
    2: (1)
    1: ()
   15: (3,2)
    9: (2,2)
   77: (5,4)
   49: (4,4)
  221: (7,6)
  169: (6,6)

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
conj[y_]:=If[Length[y]==0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]];
Table[Count[primeMS[n], _?OddQ]-Count[conj[primeMS[n]], _?EvenQ], {n, 100}]

Crossrefs

The conjugate version is A350849.

This is a hybrid of A195017 and A350941.

Positions of 0's are A350943.

A000041 = integer partitions, strict A000009.

A056239 adds up prime indices, counted by A001222, row sums of A112798.

A122111 represents conjugation using Heinz numbers.

A257991 = # of odd parts, conjugate A344616.

A257992 = # of even parts, conjugate A350847.

A316524 = alternating sum of prime indices.

The following rank partitions:

A325698: # of even parts = # of odd parts.

A349157: # of even parts = # of odd conjugate parts, counted by A277579.

A350848: # even conj parts = # odd conj parts, counted by A045931.

A350943: # of even conjugate parts = # of odd parts, counted by A277579.

A350944: # of odd parts = # of odd conjugate parts, counted by A277103.

A350945: # of even parts = # of even conjugate parts, counted by A350948.

Cf. A026424, A028260, A130780, A171966, A239241, A241638, A325700, A350947, A350949, A350950, A350951.

Keyword

sign

Author

Gus Wiseman, Jan 28 2022

Status

approved

A053250 Coefficients of the '3rd-order' mock theta function phi(q).

1, 1, 0, -1, 1, 1, -1, -1, 0, 2, 0, -2, 1, 1, -1, -2, 1, 3, -1, -2, 1, 2, -2, -3, 1, 4, 0, -4, 2, 3, -2, -4, 1, 5, -2, -5, 3, 5, -3, -5, 2, 7, -2, -7, 3, 6, -4, -8, 3, 9, -2, -9, 5, 9, -5, -10, 3, 12, -4, -12, 5, 11, -6, -13, 6, 16, -6, -15, 7, 15, -8, -17, 7, 19, -6, -20, 9, 19, -10, -22, 8, 25, -9, -25, 12, 25, -12, -27, 11, 31

Offset

0,10

References

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 55, Eq. (26.12), p. 58, Eq. (26.56).

Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 17, 31.

Links

T. D. Noe, Table of n, a(n) for n = 0..1000

Leila A. Dragonette, Some asymptotic formulas for the mock theta series of Ramanujan, Trans. Amer. Math. Soc., 72 (1952) 474-500.

John F. R. Duncan, Michael J. Griffin and Ken Ono, Proof of the Umbral Moonshine Conjecture, arXiv:1503.01472 [math.RT], 2015.

A. Folsom, K. Ono and R. C. Rhoades, Ramanujan's radial limits, 2013. - From N. J. A. Sloane, Feb 09 2013

George N. Watson, The final problem: an account of the mock theta functions, J. London Math. Soc., 11 (1936) 55-80.

Formula

Consider partitions of n into distinct odd parts. a(n) = number of them for which the largest part minus twice the number of parts is == 3 (mod 4) minus the number for which it is == 1 (mod 4).

a(n) = (-1)^n*(A027358(n)-A027357(n)). - Vladeta Jovovic, Mar 12 2006

G.f.: 1 + Sum_{k>0} x^k^2 / ((1 + x^2) (1 + x^4) ... (1 + x^(2*k))).

G.f. 1 + Sum_{n >= 0} x^(2*n+1)*Product_{k = 1..n} (x^(2*k-1) - 1) (Folsom et al.). Cf. A207569 and A215066. - Peter Bala, May 16 2017

Example

G.f. = 1 + x - x^3 + x^4 + x^5 - x^6 - x^7 + 2*x^9 - 2*x^11 + x^12 + x^13 - x^14 + ...

Maple

f:=n->q^(n^2)/mul((1+q^(2*i)), i=1..n); add(f(n), n=0..10);

Mathematica

Series[Sum[q^n^2/Product[1+q^(2k), {k, 1, n}], {n, 0, 10}], {q, 0, 100}]
a[ n_] := SeriesCoefficient[ Sum[ x^k^2 / QPochhammer[ -x^2, x^2, k], {k, 0, Sqrt@ n}], {x, 0, n}]; (* Michael Somos, Jul 09 2015 *)

Prog

(PARI) {a(n) = my(t); if(n<0, 0, t = 1 + O(x^n); polcoeff( sum(k=1, sqrtint(n), t *= x^(2*k - 1) / (1 + x^(2*k)) + O(x^(n - (k-1)^2 + 1)), 1), n))}; /* Michael Somos, Jul 16 2007 */

Crossrefs

Other '3rd-order' mock theta functions are at A000025, A053251, A053252, A053253, A053254, A053255.

Cf. A207569, A215066.

Keyword

sign,easy

Author

Dean Hickerson, Dec 19 1999

Status

approved

A345415 Table read by upward antidiagonals: Given m, n >= 1, write gcd(m,n) as d = u*m+v*n where u, v are minimal; T(m,n) = u.

0, 0, 1, 0, 0, 1, 0, 1, -1, 1, 0, 0, 0, 1, 1, 0, 1, 1, -1, -2, 1, 0, 0, -1, 0, 2, 1, 1, 0, 1, 0, 1, -1, 1, -3, 1, 0, 0, 1, 1, 0, -1, -2, 1, 1, 0, 1, -1, -1, 1, -1, 2, 3, -4, 1, 0, 0, 0, 0, -2, 0, 3, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, -1, -3, -2, -3, -5, 1, 0, 0, -1, 1, -1, 1, 0, -1, 2, -2, 4, 1, 1

Offset

1,20

Comments

The gcd is given in A003989, and v is given in A345416. Minimal means minimize u^2+v^2. We follow Maple, PARI, etc., in setting u=0 and v=1 when m=n. If we ignore the diagonal, the v table is the transpose of the u table.

Links

Table of n, a(n) for n=1..91.

Example

The gcd table (A003989) begins:
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
[1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2]
[1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1]
[1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4]
[1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1]
[1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2]
[1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1]
[1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 8]
[1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1]
[1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 5, 2]
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1]
[1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12, 1, 2, 3, 4]
...
The u table (this entry) begins:
[0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
[0, 0, -1, 1, -2, 1, -3, 1, -4, 1, -5, 1, -6, 1, -7, 1]
[0, 1, 0, -1, 2, 1, -2, 3, 1, -3, 4, 1, -4, 5, 1, -5]
[0, 0, 1, 0, -1, -1, 2, 1, -2, -2, 3, 1, -3, -3, 4, 1]
[0, 1, -1, 1, 0, -1, 3, -3, 2, 1, -2, 5, -5, 3, 1, -3]
[0, 0, 0, 1, 1, 0, -1, -1, -1, 2, 2, 1, -2, -2, -2, 3]
[0, 1, 1, -1, -2, 1, 0, -1, 4, 3, -3, -5, 2, 1, -2, 7]
[0, 0, -1, 0, 2, 1, 1, 0, -1, -1, -4, -1, 5, 2, 2, 1]
[0, 1, 0, 1, -1, 1, -3, 1, 0, -1, 5, -1, 3, -3, 2, -7]
[0, 0, 1, 1, 0, -1, -2, 1, 1, 0, -1, -1, 4, 3, -1, -3]
[0, 1, -1, -1, 1, -1, 2, 3, -4, 1, 0, -1, 6, -5, -4, 3]
[0, 0, 0, 0, -2, 0, 3, 1, 1, 1, 1, 0, -1, -1, -1, -1]
...
The v table (A345416) begins:
[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
[1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0]
[1, -1, 1, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1]
[1, 1, -1, 1, 1, 1, -1, 0, 1, 1, -1, 0, 1, 1, -1, 0]
[1, -2, 2, -1, 1, 1, -2, 2, -1, 0, 1, -2, 2, -1, 0, 1]
[1, 1, 1, -1, -1, 1, 1, 1, 1, -1, -1, 0, 1, 1, 1, -1]
[1, -3, -2, 2, 3, -1, 1, 1, -3, -2, 2, 3, -1, 0, 1, -3]
[1, 1, 3, 1, -3, -1, -1, 1, 1, 1, 3, 1, -3, -1, -1, 0]
[1, -4, 1, -2, 2, -1, 4, -1, 1, 1, -4, 1, -2, 2, -1, 4]
[1, 1, -3, -2, 1, 2, 3, -1, -1, 1, 1, 1, -3, -2, 1, 2]
[1, -5, 4, 3, -2, 2, -3, -4, 5, -1, 1, 1, -5, 4, 3, -2]
[1, 1, 1, 1, 5, 1, -5, -1, -1, -1, -1, 1, 1, 1, 1, 1]
...

Maple

mygcd:=proc(a, b) local d, s, t; d := igcdex(a, b, `s`, `t`); [a, b, d, s, t]; end;
gcd_rowu:=(m, M)->[seq(mygcd(m, n)[4], n=1..M)];
for m from 1 to 12 do lprint(gcd_rowu(m, 16)); od;

Mathematica

T[m_, n_] := Module[{u, v}, MinimalBy[{u, v} /. Solve[u^2 + v^2 <= 26 && u*m + v*n == GCD[m, n], {u, v}, Integers], #.#&][[1, 1]]];
Table[T[m - n + 1, n], {m, 1, 13}, {n, 1, m}] // Flatten (* Jean-François Alcover, Mar 27 2023 *)

Crossrefs

Cf. A003989, A050873, A345416, A345417, A345418.

Cf. also A345872, A345873.

Keyword

sign,tabl

Author

N. J. A. Sloane, Jun 19 2021

Status

approved

A064334 Triangle composed of generalized Catalan numbers.

1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, -1, 1, 1, 1, -2, 5, -2, 1, 1, 1, 6, -25, 13, -3, 1, 1, 1, -18, 141, -98, 25, -4, 1, 1, 1, 57, -849, 826, -251, 41, -5, 1, 1, 1, -186, 5349, -7448, 2817, -514, 61, -6, 1, 1, 1, 622, -34825, 70309, -33843, 7206, -917, 85, -7, 1, 1

Offset

0,17

Comments

The sequence for column m (m >= 1) (without leading zeros and the first 1) appears in the Derrida et al. 1992 reference as Z_{N}=:Y_{N}(N+1), N >=0, for (unphysical) alpha = -m, beta = 1 (or alpha = 1, beta = -m). In the Derrida et al. 1993 reference the formula in eq. (39) gives Z_{N}(alpha,beta)/(alpha*beta)^N for N>=1. See also Liggett reference, proposition 3.19, p. 269, with lambda for alpha and rho for 1-beta.

References

T. M. Liggett, Stochastic Interacting Systems: Contact, Voter and Exclusion Processes, Springer, 1999, p. 269.

Links

G. C. Greubel, Rows n =0..100 of triangle, flattened

B. Derrida, E. Domany and D. Mukamel, An exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 1992, 667-687; eqs. (20), (21), p. 672.

B. Derrida, M. R. Evans, V. Hakim and V. Pasquier, Exact solution of a 1D asymmetric exclusion model using a matrix formulation, J. Phys. A 26, 1993, 1493-1517; eq. (39), p. 1501, also appendix A1, (A12) p. 1513.

Formula

G.f. for column m: (x^m)/(1-x*c(-m*x))= (x^m)*((m+1)+m*x*c(-m*x))/((m+1)-x), m>0, with the g.f. c(x) of Catalan numbers A000108.

T(n, m) = Sum_{k=0..n-m-1} (n-m-k)*binomial(n-m-1+k, k)*(-m)^k/(n-m), with T(n,0) = T(n,n)=1.

T(n,m) = (1/(1+m))^(n-m)*(1 + m*Sum_{k=0..n-m-1} C(k)*(-m*(m+1))^k ), n-m >= 1, T(n, n) = T(n,0) =1, T(n, m)=0 if n<m, with C(k)=A000108(k) (Catalan).

T(n, k) = hypergeometric([1-n+k, n-k], [-n+k], -k) if k<n else 1. - Peter Luschny, Nov 30 2014

Example

Triangle starts:
  1;
  1,  1;
  1,  1,  1;
  1,  0,  1,  1;
  1,  1, -1,  1, 1;
  1, -2,  5, -2, 1, 1; ...

Mathematica

Table[If[k==0, 1, If[k==n, 1, Sum[(n-k-j)*Binomial[n-k-1+j, j]*(-k)^j/(n -k), {j, 0, n-k-1}]]], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, May 04 2019 *)

Prog

(Sage)
def T(n, k):
    return hypergeometric([1-n, n], [-n], -k) if n>0 else 1
for n in (0..10):
    print([simplify(T(n-k, k)) for k in (0..n)]) # Peter Luschny, Nov 30 2014
(PARI) {T(n, k) = if(k==0, 1, if(k==n, 1, sum(j=0, n-k-1, (n-k-j)* binomial(n-k-1+j, j)*(-k)^j/(n-k))))}; \\ G. C. Greubel, May 04 2019
(Magma) [[k eq 0 select 1 else k eq n select 1 else (&+[(n-k-j)* Binomial(n-k-1+j, j)*(-k)^j/(n-k): j in [0..n-k-1]]): k in [0..n]]: n in [0..12]]; // G. C. Greubel, May 04 2019

Crossrefs

The unsigned column sequences (without leading zeros) are A000012, A064310-11, A064325-33 for m=0..11, respectively. Row sums (signed) give A064338. Row sums (unsigned) give A064339.

Cf. A064062.

Keyword

sign,easy,tabl

Author

Wolfdieter Lang, Sep 21 2001

Status

approved

A098593 A triangle of Krawtchouk coefficients.

1, 1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -2, -2, -2, 1, 1, -3, -2, -2, -3, 1, 1, -4, -1, 0, -1, -4, 1, 1, -5, 1, 3, 3, 1, -5, 1, 1, -6, 4, 6, 6, 6, 4, -6, 1, 1, -7, 8, 8, 6, 6, 8, 8, -7, 1, 1, -8, 13, 8, 2, 0, 2, 8, 13, -8, 1, 1, -9, 19, 5, -6, -10, -10, -6, 5, 19, -9, 1, 1, -10, 26, -2, -17, -20, -20, -20, -17, -2, 26, -10, 1, 1, -11, 34, -14, -29, -25

Offset

0,12

Comments

Row sums are A009545(n+1), with e.g.f. exp(x)(cos(x)+sin(x)). Diagonal sums are A077948.

The rows are the diagonals of the Krawtchouk matrices. Coincides with the Riordan array (1/(1-x),(1-2x)/(1-x)). - Paul Barry, Sep 24 2004

Corresponds to Pascal-(1,-2,1) array, read by antidiagonals. The Pascal-(1,-2,1) array has n-th row generated by (1-2x)^n/(1-x)^(n+1). - Paul Barry, Sep 24 2004

A modified version (different signs) of this triangle is given by T(n,k) = Sum_{j=0..n} C(n-k,j)*C(k,j)*cos(Pi*(k-j)). - Paul Barry, Jun 14 2007

References

P. Feinsilver and J. Kocik, Krawtchouk matrices from classical and quantum walks, Contemporary Mathematics, 287 2001, pp. 83-96.

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Paul Barry, A note on Krawtchouk Polynomials and Riordan Arrays, JIS 11 (2008) 08.2.2

P. Feinsilver and J. Kocik, Krawtchouk polynomials and Krawtchouk matrices, arxiv:quant-ph/0702073, 2007.

Formula

T(n, k) = Sum_{i=0..k} binomial(n-k, k-i)*binomial(k, i)*(-1)^(k-i), k<=n.

T(n, k) = T(n-1, k) + T(n-1, k-1) - 2*T(n-2, k-1) (n>0). - Paul Barry, Sep 24 2004

T(n, k) = [k<=n]*Hypergeometric2F1(-k,k-n;1;-1). - Paul Barry, Jan 24 2011

E.g.f. for the n-th subdiagonal: exp(x)*P(n,x), where P(n,x) is the polynomial Sum_{k = 0..n} (-1)^k*binomial(n,k)* x^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 - 2*x + x^2/2) = 1 - x - 2*x^2/2! - 2*x^3/3! - x^4/4! + x^5/5! + .... - Peter Bala, Mar 05 2017

Example

Rows begin {1}, {1,1}, {1,0,1}, {1,-1,-1,1}, {1,-2,-2,-2,1}, ...
From Paul Barry, Oct 05 2010: (Start)
Triangle begins
  1,
  1,  1,
  1,  0,  1,
  1, -1, -1,  1,
  1, -2, -2, -2,  1,
  1, -3, -2, -2, -3,  1,
  1, -4, -1,  0, -1, -4,  1,
  1, -5,  1,  3,  3,  1, -5,  1,
  1, -6,  4,  6,  6,  6,  4, -6,  1,
  1, -7,  8,  8,  6,  6,  8,  8, -7,  1,
  1, -8, 13,  8,  2,  0,  2,  8, 13, -8,  1
Production matrix (related to large Schroeder numbers A006318) begins
  1,     1,
  0,    -1,     1,
  0,    -2,    -1,    1,
  0,    -6,    -2,   -1,   1,
  0,   -22,    -6,   -2,  -1,   1,
  0,   -90,   -22,   -6,  -2,  -1,  1,
  0,  -394,   -90,  -22,  -6,  -2, -1,  1,
  0, -1806,  -394,  -90, -22,  -6, -2, -1,  1,
  0, -8558, -1806, -394, -90, -22, -6, -2, -1, 1
Production matrix of inverse is
    -1,   1,
    -2,   1,  1,
    -4,   2,  1,  1,
    -8,   4,  2,  1,  1,
   -16,   8,  4,  2,  1, 1,
   -32,  16,  8,  4,  2, 1, 1,
   -64,  32, 16,  8,  4, 2, 1, 1,
  -128,  64, 32, 16,  8, 4, 2, 1, 1,
  -256, 128, 64, 32, 16, 8, 4, 2, 1, 1 (End)

Mathematica

T[n_, k_] := Sum[Binomial[n - k, k - j]*Binomial[k, j]*(-1)^(k - j), {j, 0, n}]; Table[T[n, k], {n, 0, 49}, {k, 0, n}] // Flatten (* G. C. Greubel, Oct 15 2017 *)

Prog

(PARI) for(n=0, 10, for(k=0, n, print1(sum(i=0, k, binomial(n-k, k-i) *binomial(k, i)*(-1)^(k-i)), ", "))) \\ G. C. Greubel, Oct 15 2017

Crossrefs

Cf. Pascal (1,m,1) array: A123562 (m = -3), A000012 (m = -1), A007318 (m = 0), A008288 (m = 1), A081577 (m = 2), A081578 (m = 3), A081579 (m = 4), A081580 (m = 5), A081581 (m = 6), A081582 (m = 7), A143683 (m = 8).

Keyword

easy,sign,tabl

Author

Paul Barry, Sep 17 2004

Status

approved