A214292 Triangle read by rows: T(n,k) = T(n-1,k-1) + T(n-1,k), 0 < k < n with T(n,0) = n and T(n,n) = -n.

0, 1, -1, 2, 0, -2, 3, 2, -2, -3, 4, 5, 0, -5, -4, 5, 9, 5, -5, -9, -5, 6, 14, 14, 0, -14, -14, -6, 7, 20, 28, 14, -14, -28, -20, -7, 8, 27, 48, 42, 0, -42, -48, -27, -8, 9, 35, 75, 90, 42, -42, -90, -75, -35, -9, 10, 44, 110, 165, 132, 0, -132, -165, -110, -44, -10

Offset

0,4

Comments

T(n,k) = A007318(n+1,k+1) - A007318(n+1,k), 0<=k<=n, i.e. first differences of rows in Pascal's triangle;

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

row sums and central terms equal 0, cf. A000004;

sum of positive elements of n-th row = A014495(n+1);

T(n,0) = n;

T(n,1) = A000096(n-2) for n > 1; T(n,1) = - A080956(n) for n > 0;

T(n,2) = A005586(n-4) for n > 3; T(n,2) = A129936(n-2);

T(n,3) = A005587(n-6) for n > 5;

T(n,4) = A005557(n-9) for n > 8;

T(n,5) = A064059(n-11) for n > 10;

T(n,6) = A064061(n-13) for n > 12;

T(n,7) = A124087(n) for n > 14;

T(n,8) = A124088(n) for n > 16;

T(2*n+1,n) = T(2*n+2,n) = A000108(n+1), Catalan numbers;

T(2*n+3,n) = A000245(n+2);

T(2*n+4,n) = A002057(n+1);

T(2*n+5,n) = A000344(n+3);

T(2*n+6,n) = A003517(n+3);

T(2*n+7,n) = A000588(n+4);

T(2*n+8,n) = A003518(n+4);

T(2*n+9,n) = A001392(n+5);

T(2*n+10,n) = A003519(n+5);

T(2*n+11,n) = A000589(n+6);

T(2*n+12,n) = A090749(n+6);

T(2*n+13,n) = A000590(n+7).

Links

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

Index entries for triangles and arrays related to Pascal's triangle

Example

The triangle begins:
.   0:                              0
.   1:                            1   -1
.   2:                          2   0   -2
.   3:                       3    2   -2   -3
.   4:                     4    5   0   -5   -4
.   5:                  5    9    5   -5   -9   -5
.   6:                6   14   14   0  -14  -14   -6
.   7:             7   20   28   14  -14  -28  -20   -7
.   8:           8   27   48   42   0  -42  -48  -27   -8
.   9:        9   35   75   90   42  -42  -90  -75  -35   -9
.  10:     10   44  110  165  132   0 -132 -165 -110  -44  -10
.  11:  11   54  154  275  297  132 -132 -297 -275 -154  -54  -11  .

Mathematica

row[n_] := Table[Binomial[n, k], {k, 0, n}] // Differences;
T[n_, k_] := row[n + 1][[k + 1]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 31 2018 *)

Prog

(Haskell)
a214292 n k = a214292_tabl !! n !! k
a214292_row n = a214292_tabl !! n
a214292_tabl = map diff $ tail a007318_tabl
   where diff row = zipWith (-) (tail row) row

Crossrefs

Cf. A007318, A000004, A000096, A000108, A000245, A000344, A000588, A000589, A000590, A001392, A002057, A003517, A003518, A003519, A005557, A005586, A005587, A008313, A014495, A064059, A064061, A080956, A090749, A097808, A112467, A124087, A124088, A129936, A259525.

Keyword

sign,tabl

Author

Reinhard Zumkeller, Jul 12 2012

Status

approved

A112467 Riordan array ((1-2x)/(1-x), x/(1-x)).

1, -1, 1, -1, 0, 1, -1, -1, 1, 1, -1, -2, 0, 2, 1, -1, -3, -2, 2, 3, 1, -1, -4, -5, 0, 5, 4, 1, -1, -5, -9, -5, 5, 9, 5, 1, -1, -6, -14, -14, 0, 14, 14, 6, 1, -1, -7, -20, -28, -14, 14, 28, 20, 7, 1, -1, -8, -27, -48, -42, 0, 42, 48, 27, 8, 1, -1, -9, -35, -75, -90, -42, 42, 90, 75, 35, 9, 1, -1, -10, -44, -110, -165, -132, 0, 132, 165, 110

Offset

0,12

Comments

Row sums are A000007. Diagonal sums are -F(n-2). Inverse is A112468. T(2n,n)=0.

(-1,1)-Pascal triangle. - Philippe Deléham, Aug 07 2006

Apart from initial term, same as A008482. - Philippe Deléham, Nov 07 2006

Each column equals the cumulative sum of the previous column. - Mats Granvik, Mar 15 2010

Reading along antidiagonals generates in essence rows of A192174. - Paul Curtz, Oct 02 2011

Triangle T(n,k), read by rows, given by (-1,2,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 01 2011

Links

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

Elena Barcucci, Antonio Bernini, Stefano Bilotta, and Renzo Pinzani, Restricting Dyck Paths and 312-avoiding Permutations, arXiv:2307.02837 [math.CO], 2023. Mentions this sequence.

Paul Barry, A Note on Riordan Arrays with Catalan Halves, arXiv:1912.01124 [math.CO], 2019.

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

D. Foata and G.-N. Han, The doubloon polynomial triangle, Ram. J. 23 (2010), 107-126.

Jack Ramsay, On Arithmetical Triangles, The Pulse of Long Island, June 1965. [Mentions application to design of antenna arrays. Annotated scan.]

Formula

Number triangle T(n, k) = binomial(n, n-k) - 2*binomial(n-1, n-k-1).

Sum_{k=0..n} T(n, k)*x^k = (x-1)*(x+1)^(n-1). - Philippe Deléham, Oct 03 2005

T(n,k) = ((2*k-n)/n)*binomial(n, k), with T(0,0)=1. - Roger L. Bagula, Feb 16 2009; modified by G. C. Greubel, Dec 04 2019

T(n,k) = T(n-1,k-1) + T(n-1,k) with T(0,0)=1, T(1,0)=-1, T(n,k)=0 for k>n or for n<0. - Philippe Deléham, Nov 01 2011

G.f.: (1-2x)/(1-(1+y)*x). - Philippe Deléham, Dec 15 2011

Sum_{k=0..n} T(n,k)*x^k = A000007(n), A133494(n), A081294(n), A005053(n), A067411(n), A199661(n), A083233(n) for x = 1, 2, 3, 4, 5, 6, 7, respectively. - Philippe Deléham, Dec 15 2011

exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(-1 - x + x^2/2! + x^3/3!) = -1 - 2*x - 2*x^2/2! + 5*x^4/4! + 14*x^5/5! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 21 2014

Sum_{k=0..n} T(n,k) = 0^n = A000007(n). - G. C. Greubel, Dec 04 2019

Example

Triangle starts:
    1;
   -1,  1;
   -1,  0,   1;
   -1, -1,   1,   1;
   -1, -2,   0,   2,   1;
   -1, -3,  -2,   2,   3,   1;
   -1, -4,  -5,   0,   5,   4,  1;
   -1, -5,  -9,  -5,   5,   9,  5,  1;
   -1, -6, -14, -14,   0,  14, 14,  6,  1;
   -1, -7, -20, -28, -14,  14, 28, 20,  7,  1;
   -1, -8, -27, -48, -42,   0, 42, 48, 27,  8, 1;
   -1, -9, -35, -75, -90, -42, 42, 90, 75, 35, 9, 1;
  ...
From Paul Barry, Apr 08 2011: (Start)
Production matrix begins:
   1,  1,
  -2, -1,  1,
   2,  0, -1,  1,
  -2,  0,  0, -1,  1,
   2,  0,  0,  0, -1,  1,
  -2,  0,  0,  0,  0, -1,  1,
   2,  0,  0,  0,  0,  0, -1,  1
  ... (End)

Maple

seq(seq( `if`(n=0, 1, (2*k-n)*binomial(n, k)/n), k=0..n), n=0..10); # G. C. Greubel, Dec 04 2019

Mathematica

T[n_, k_]= If[n==0, 1, ((2*k-n)/n)*Binomial[n, k]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Roger L. Bagula, Feb 16 2009; modified by G. C. Greubel, Dec 04 2019 *)

Prog

(PARI) T(n, k) = if(n==0, 1, (2*k-n)*binomial(n, k)/n ); \\ G. C. Greubel, Dec 04 2019
(Magma) [n eq 0 select 1 else (2*k-n)*Binomial(n, k)/n: k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 04 2019
(Sage)
def T(n, k):
    if (n==0): return 1
    else: return (2*k-n)*binomial(n, k)/n
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 04 2019

Crossrefs

Same first 3 rows as in A054525.

Cf. A008482, A037012, A080232, A112466, A112467, A174293, A174294, A174295, A174296, A174297.

Keyword

easy,sign,tabl

Author

Paul Barry, Sep 06 2005

Status

approved

A097808 Riordan array ((1+2x)/(1+x)^2, 1/(1+x)) read by rows.

1, 0, 1, -1, -1, 1, 2, 0, -2, 1, -3, 2, 2, -3, 1, 4, -5, 0, 5, -4, 1, -5, 9, -5, -5, 9, -5, 1, 6, -14, 14, 0, -14, 14, -6, 1, -7, 20, -28, 14, 14, -28, 20, -7, 1, 8, -27, 48, -42, 0, 42, -48, 27, -8, 1, -9, 35, -75, 90, -42, -42, 90, -75, 35, -9, 1, 10, -44, 110, -165, 132, 0, -132, 165, -110, 44, -10, 1

Offset

0,7

Comments

Inverse of A059260. Row sums are inverse binomial transform of A040000, with g.f. (1+2x)/(1+x). Diagonal sums are (-1)^n(1-Fib(n)). A097808=B^(-1)*A097806, where B is the binomial matrix. B*A097808*B^(-1) is the inverse of A097805.

Links

Robert Israel, Table of n, a(n) for n = 0..10010 (rows 0 to 140, flattened)

Formula

Columns have g.f. (1+2x)/(1+x)^2(x/(1+x))^k.

T(n,k)=T(n-1,k-1)-3*T(n-1,k)+2*T(n-2,k-1)-3*T(n-2,k)+T(n-3,k-1)-T(n-3,k), T(0,0)=T(1,1)=T(2,2)=1, T(1,0)=0, T(2,0)=T(2,1)=-1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Jan 12 2014

T(0,0)=1, T(n,0)=(-1)^(n-1)*(n-1) for n>0, T(n,n)=1, T(n,k)=T(n-1,k-1)-T(n-1,k) for 0<k<n. - Philippe Deléham, Jan 12 2014

Example

Rows begin
1;
0, 1;
-1, -1, 1;
2, 0, -2, 1;
-3, 2, 2, -3, 1;
4, -5, 0, 5, -4, 1;
-5, 9, -5, -5, 9, -5, 1;
6, -14, 14, 0, -14, 14, -6, 1;
-7, 20, -28, 14, 14, -28, 20, -7, 1;
8, -27, 48, -42, 0, 42, -48, 27, -8, 1;

Maple

T:= proc(n, k) option remember;
if k < 0 or k > n then return 0 fi;
procname (n-1, k-1)-3*procname(n-1, k)+2*procname(n-2, k-1)-3*procname(n-2, k)+
procname(n-3, k-1)-procname(n-3, k)
end proc:
T(0, 0):= 1: T(1, 1):= 1: T(2, 2):= 1:
T(1, 0):= 0: T(2, 0):= -1: T(2, 1):= -1:
seq(seq(T(n, k), k=0..n), n=0..12); # Robert Israel, Jul 16 2019

Mathematica

(* The function RiordanArray is defined in A256893. *)
RiordanArray[(1 + 2 #)/(1 + #)^2&, #/(1 + #)&, 12] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

Keyword

easy,sign,tabl

Author

Paul Barry, Aug 25 2004

Status

approved

A153174 Coefficients of the eighth-order mock theta function U_1(q).

0, 1, 0, -1, 1, 2, -1, -2, 1, 3, -1, -4, 2, 5, -2, -6, 3, 8, -4, -9, 4, 11, -5, -14, 7, 17, -7, -20, 9, 24, -11, -28, 12, 33, -15, -39, 18, 46, -20, -53, 24, 62, -28, -72, 32, 83, -37, -96, 43, 110, -48, -126, 56, 145, -65, -165, 72, 188, -83, -214, 95, 243

Offset

0,6

Links

Table of n, a(n) for n=1..62.

B. Gordon and R. J. McIntosh, Some eighth order mock theta functions, J. London Math. Soc. 62 (2000), 321-335.

Formula

G.f: Sum_{n >= 0} q^((n+1)^2)(1+q)(1+q^3)...(1+q^(2n-1))/((1+q^2)(1+q^6)...(1+q^(4n+2))).

Prog

(PARI) lista(nn) = {my(q = qq + O(qq^nn)); gf = sum(n = 0, nn, q^((n+1)^2) * prod(k = 1, n, 1 + q^(2*k-1)) / prod(k = 0, n, 1 + q^(4*k+2))); for (i=0, nn-1, print1(polcoeff(gf, i), ", "); ); } \\ Michel Marcus, Jun 18 2013

Crossrefs

Other '8th-order' mock theta functions are at A153148, A153149, A153155, A153156, A153172, A153176, A153178.

Keyword

sign

Author

Jeremy Lovejoy, Dec 20 2008

Extensions

More terms from Michel Marcus, Feb 23 2015

Status

approved

A058511 McKay-Thompson series of class 15D for the Monster group.

1, -2, -1, 2, 1, 4, -6, -2, 2, 0, 10, -14, -5, 8, 4, 20, -28, -10, 14, 4, 39, -56, -20, 28, 10, 72, -100, -34, 46, 16, 128, -176, -61, 86, 30, 216, -294, -100, 134, 44, 355, -484, -165, 226, 79, 568, -770, -260, 350, 116, 894, -1208, -408, 552, 188, 1376, -1848, -620, 830, 276, 2087, -2800, -940

Offset

0,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.

D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).

J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.

Index entries for McKay-Thompson series for Monster simple group

Formula

Expansion of q^(1/3) * (eta(q) / eta(q^5))^2 in powers of q.

Euler transform of period 5 sequence [ -2, -2, -2, -2, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (45 t)) = 5 / f(t) where q = exp(2 Pi i t).

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u - v^2) * (v - u^2) + 4*u*v.

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 + u*w + w^2 - v^2 * (u + w) - 5*v.

Example

G.f. = 1 - 2*x - x^2 + 2*x^3 + x^4 + 4*x^5 - 6*x^6 - 2*x^7 + 2*x^8 + ...
T15D = 1/q - 2*q^2 - q^5 + 2*q^8 + q^11 + 4*q^14 - 6*q^17 - 2*q^20 + 2*q^23 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] / QPochhammer[ x^5])^2, {x, 0, n}]; (* Michael Somos, Aug 26 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^5 + A))^2, n))}; /* Michael Somos, Dec 17 2010 */

Crossrefs

Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.

Keyword

sign

Author

N. J. A. Sloane, Nov 27 2000

Status

approved

A112466 Riordan array ((1+2x)/(1+x), x/(1+x)).

1, 1, 1, -1, 0, 1, 1, -1, -1, 1, -1, 2, 0, -2, 1, 1, -3, 2, 2, -3, 1, -1, 4, -5, 0, 5, -4, 1, 1, -5, 9, -5, -5, 9, -5, 1, -1, 6, -14, 14, 0, -14, 14, -6, 1, 1, -7, 20, -28, 14, 14, -28, 20, -7, 1, -1, 8, -27, 48, -42, 0, 42, -48, 27, -8, 1, 1, -9, 35, -75, 90, -42, -42, 90, -75, 35, -9, 1, -1, 10, -44, 110, -165, 132, 0, -132, 165, -110, 44

Offset

0,12

Comments

Row sums are (1,2,0,0,0,...).

Inverse is A112465.

Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 07 2006; corrected by Philippe Deléham, Dec 11 2008

Equals A097808 when the first column is removed. - Georg Fischer, Jul 26 2023

Links

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)

Paul Barry, A Note on Riordan Arrays with Catalan Halves, arXiv:1912.01124 [math.CO], 2019.

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

Formula

Number triangle T(n,k) = (-1)^(n-k)*(C(n, n-k) - 2*C(n-1, n-k-1)).

Sum_{k=0..floor(n/2)} T(n-k,k) = (-1)^(n+1)*Fibonacci(n-2).

T(2n,n) = 0.

Sum_{k=0..n} T(n,k)*x^k = (x+1)*(x-1)^(n-1), for n >= 1. - Philippe Deléham, Oct 03 2005

T(0,0)=T(1,0)=T(1,1)=1, T(n,k)=0 if n < 0 or if n < k, T(n,k) = T(n-1,k-1) - T(n-1,k) for n > 1. - Philippe Deléham, Nov 26 2006

G.f.: (1+2*x)/(1+x-x*y). - R. J. Mathar, Aug 11 2015

Example

Triangle starts
   1;
   1,  1;
  -1,  0,  1;
   1, -1, -1,  1;
  -1,  2,  0, -2,  1;
   1, -3,  2,  2, -3,  1;
  -1,  4, -5,  0,  5, -4,  1;
From Paul Barry, Apr 08 2011: (Start)
Production matrix begins
   1,  1;
  -2, -1,  1;
   2,  0, -1,  1;
  -2,  0,  0, -1,  1;
   2,  0,  0,  0, -1,  1;
  -2,  0,  0,  0,  0, -1,  1;
   2,  0,  0,  0,  0,  0, -1,  1; (End)

Maple

seq(seq( (-1)^(n-k)*(2*binomial(n-1, k-1)-binomial(n, k)), k=0..n), n=0..10); # G. C. Greubel, Feb 19 2020

Mathematica

{1}~Join~Table[(Binomial[n, n - k] - 2 Binomial[n - 1, n - k - 1])*(-1)^(n - k), {n, 12}, {k, 0, n}] // Flatten (* Michael De Vlieger, Feb 18 2020 *)

Prog

(PARI) T(n, k) = (-1)^(n-k)*(binomial(n, n-k) - 2*binomial(n-1, n-k-1)); \\ Michel Marcus, Feb 19 2020

Crossrefs

Cf. A008482, A037012, A097808, A112467.

Keyword

easy,sign,tabl

Author

Paul Barry, Sep 06 2005

Status

approved

A197419 Triangle with the numerator of the coefficient [x^k] of the second order Bernoulli polynomial B_n^(2)(x) in row n, column 0<=k<=n.

1, -1, 1, 5, -2, 1, -1, 5, -3, 1, 1, -2, 5, -4, 1, 1, 1, -5, 25, -5, 1, -5, 1, 3, -10, 25, -6, 1, -1, -5, 7, 7, -35, 35, -7, 1, 7, -4, -10, 28, 7, -28, 70, -8, 1, 3, 21, -6, -10, 21, 63, -42, 30, -9, 1, -15, 3, 21, -20, -25, 42, 21, -60, 75, -10, 1, -5, -15, 33, 77, -55, -55, 77, 33, -165, 275, -11, 1, 7601, -10, -45, 66, 231, -132, -110, 132, 99, -110, 55, -12, 1

Offset

0,4

Comments

The a-th order Bernoulli polynomials are defined via the exponential generating function (t/(exp t -1))^a*exp(x*t) = sum_{n>=0} B_n^(a)(x) * t^n/n!. The current triangular array shows the coefficient [x^k] of B_n^(2)(x), i.e. the expansion coefficients in rising powers of the polynomial of x with a=2.

P(n,x) = 2*sum(m=0..n-1, binomial(n,m)*sum(k=1..n-m, stirling2(n-m,k) * stirling1(2+k,2)/((k+1)*(k+2))))*x^m+x^n. - Vladimir Kruchinin, Oct 23 2011]

Links

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

R. Dere, Y. Simsek, Bernoulli type polynomials on Umbral Algebra, arXiv:1110.1484 [math.CA]

V. Kruchinin, D. Kruchinin, Application of a composition of generating functions for obtaining explicit formulas of polynomials, arXiv: 1211.0099

Formula

T(n,m) = sum(2*C(n,m)*sum(k=1..n-m, stirling2(n-m,k)*stirling1(2+k,2)/ ((k+1)*(2+k)))), m<n, T(n,n)=1. - Vladimir Kruchinin, Oct 23 2011

Example

The table of the coefficients is
1;
-1,1;
5/6,-2,1;     5/6-2x+x^2
-1/2,5/2,-3,1;   -1/2+5x/2-3x^2+x^3
1/10,-2,5,-4,1;
1/6,1/2,-5,25/3,-5,1;
-5/42,1,3/2,-10,25/2,-6,1;
-1/6,-5/6,7/2,7/2,-35/2,35/2,-7,1;
7/30,-4/3,-10/3,28/3,7,-28,70/3,-8,1;
3/10,21/10,-6,-10,21,63/5,-42,30,-9,1;
-15/22,3,21/2,-20,-25,42,21,-60,75/2,-10,1;
-5/6,-15/2,33/2,77/2,-55,-55,77,33,-165/2,275/6,-11,1;
7601/2730,-10,-45,66,231/2,-132,-110,132,99/2,-110,55,-12,1;

Maple

A197419 := proc(n, k)
        local a, Bt, Bnx, o , t, x;
        a := 2 ;
        Bt := (t/(exp(t)-1))^a*exp(x*t) ;
        Bnx := n!*coeftayl(Bt, t=0, n) ;
        coeftayl(Bnx, x=0, k) ;
        numer(%) ;
end proc:
seq(seq(A197419(n, k), k=0..n), n=0..4) ; # print row by row

Mathematica

t[n_, m_] := If [n == m, 1, 2*Binomial[n, m]*Sum[StirlingS2[n-m, k]*StirlingS1[2+k, 2]/((k+1)*(2+k)), {k, 1, n-m}]]; Table[t[n, m] // Numerator, {n, 0, 12}, {m, 0, n}] // Flatten (* Jean-François Alcover, Dec 12 2013, after Vladimir Kruchinin *)

Prog

(Maxima) T(n, m):=num(if n=m then 1 else 2*binomial(n, m)* sum(stirling2(n-m, k) *stirling1(2+k, 2)/ ((k+1)*(2+k)), k, 1, n-m)); [From Vladimir Kruchinin, Oct 23 2011]

Crossrefs

Cf. A197420 (denominator), A100616, A100615 (column k=0).

Keyword

sign,tabl,frac

Author

R. J. Mathar, Oct 14 2011

Status

approved

A344503 a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)^2*hypergeom([(k-n)/2, (k-n+1)/2], [k+2], 4).

1, 0, -1, 3, 0, -5, 15, 0, -28, 84, 0, -165, 495, 0, -1001, 3003, 0, -6188, 18564, 0, -38760, 116280, 0, -245157, 735471, 0, -1562275, 4686825, 0, -10015005, 30045015, 0, -64512240, 193536720, 0, -417225900, 1251677700, 0, -2707475148, 8122425444, 0, -17620076360

Offset

0,4

Comments

Inverse binomial convolution of the Motzkin numbers.

Links

Table of n, a(n) for n=1..42.

Formula

a(3*n) = binomial(3*n, n) (A005809).

a(3*n - 1) = -binomial(3*n - 1, n - 1) (A025174).

a(3*n - 2) = 0.

Conjecture D-finite with recurrence -18*(2*n+1) *(2*n-1) *(n+1) *a(n) +2*(-36*n^3+554*n^2-1128*n+27) *a(n-1) +6*(-12*n^3-188*n^2+1235*n-1618) *a(n-2) +9*(54*n^3-27*n^2-183*n+320) *a(n-3) +54*(n-3) *(9*n^2-125*n+75) *a(n-4) +81 *(n-3) *(n-4) *(6*n+127) *a(n-5)=0. - R. J. Mathar, Nov 02 2021

Maple

a := n -> add((-1)^(n - k)*binomial(n, k)^2*hypergeom([(k-n)/2, (k-n+1)/2], [k+2], 4), k = 0..n): seq(simplify(a(n)), n = 0..41);

Crossrefs

Cf. A064189 (Motzkin numbers), A005809, A025174, A344502.

Keyword

sign

Author

Peter Luschny, May 23 2021

Status

approved

A126595 Triangle read by rows: T(0,0)=1; for n>=1, 0<=k<=n, T(n,k) is the coefficient of x^k in the characteristic polynomial (-x)^n+... of the n X n matrix M(n)S(n), where M(n) is the n X n matrix with 0's on the diagonal and 1's elsewhere and S(n) is the n X n matrix whose (i,j) term is 0 for j=i, (-1)^(i+j) for i>j and (-1)^(i+j+1) for i<j.

1, 0, -1, -1, 0, 1, 0, -3, 0, -1, -3, 0, 2, 0, 1, 0, -5, 0, -10, 0, -1, -5, 0, -5, 0, 9, 0, 1, 0, -7, 0, -35, 0, -21, 0, -1, -7, 0, -28, 0, 14, 0, 20, 0, 1, 0, -9, 0, -84, 0, -126, 0, -36, 0, -1, -9, 0, -75, 0, -42, 0, 90, 0, 35, 0, 1, 0, -11, 0, -165, 0, -462, 0, -330, 0, -55, 0, -1, -11, 0, -154, 0, -297, 0, 132, 0, 275, 0, 54, 0, 1, 0, -13, 0

Offset

0,8

Comments

Sum of terms in row 2n (n>=1) is 0. Sum of the absolute values of the terms in row 2n is C(2n,n) (A000984). All terms in row 2n-1 are nonpositive. Their sum is -4^(n-1). M(2n-1)S(2n-1)=-S(2n-1)

Links

Table of n, a(n) for n=1..94.

Example

M(4)=[0,1,1,1/1,0,1,1/1,1,0,1/1,1,1,0], S(4)=[0,1,-1,1/-1,0,1,-1/1,-1,0,1/-1,1,-1,0], M(4)S(4)=[ -1,0,0,0/0,1,-2,2/-2,2,-1,0/0,0,0,1]; char. poly. of M(4)S(4) is x^4 + 2x^2 - 3, yielding row 4 of the triangle: -3,0,2,0,1.
Triangle starts:
1;
0,-1;
-1,0,1;
0,-3,0,-1;
-3,0,2,0,1;
0,-5,0,-10,0,-1

Maple

with(linalg): m:=proc(i, j) if i=j then 0 else 1 fi end: s:=proc(i, j) if i=j then 0 elif i>j then (-1)^(i+j) else (-1)^(i+j+1) fi end: for n from 1 to 14 do f[n]:=(-1)^n*sort(expand(charpoly(multiply(matrix(n, n, m), matrix(n, n, s)), x))) od: 1; for n from 1 to 14 do seq(coeff(f[n], x, j), j=0..n) od; # yields sequence in triangular form

Crossrefs

Cf. A000984.

Keyword

sign,tabl

Author

Roger L. Bagula, Jan 01 2007

Extensions

Edited by N. J. A. Sloane, Jan 07 2006

Status

approved

A099039 Riordan array (1,c(-x)), where c(x) = g.f. of Catalan numbers.

1, 0, 1, 0, -1, 1, 0, 2, -2, 1, 0, -5, 5, -3, 1, 0, 14, -14, 9, -4, 1, 0, -42, 42, -28, 14, -5, 1, 0, 132, -132, 90, -48, 20, -6, 1, 0, -429, 429, -297, 165, -75, 27, -7, 1, 0, 1430, -1430, 1001, -572, 275, -110, 35, -8, 1, 0, -4862, 4862, -3432, 2002, -1001, 429, -154, 44, -9, 1, 0, 16796, -16796, 11934, -7072, 3640, -1638

Offset

0,8

Comments

Row sums are generalized Catalan numbers A064310. Diagonal sums are 0^n+(-1)^n*A030238(n-2). Inverse is A026729, as number triangle. Columns have g.f. (xc(-x))^k=((sqrt(1+4x)-1)/2)^k.

Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, ... ] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... ] where DELTA is the operator defined in A084938. - Philippe Deléham, May 31 2005

Links

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

George Beck and Karl Dilcher, A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence, arXiv:2106.10400 [math.CO], 2021. See (2.10) p. 6.

F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999), 73-112.

D. Callan, A recursive bijective approach to counting permutations containing 3-letter patterns, arXiv:math/0211380 [math.CO], 2002.

E. Deutsch, Dyck path enumeration, Discrete Math., 204, 1999, 167-202.

R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.

A. Robertson, D. Saracino and D. Zeilberger, Refined restricted permutations, arXiv:math/0203033 [math.CO], 2002.

L. W. Shapiro, S. Getu, Wen-Jin Woan and L. C. Woodson, The Riordan Group, Discrete Appl. Maths. 34 (1991) 229-239.

Formula

T(n, k) = (-1)^(n+k)*binomial(2*n-k-1, n-k)*k/n for 0 <= k <= n with n > 0; T(0, 0) = 1; T(0, k) = 0 if k > 0. - Philippe Deléham, May 31 2005

Example

Rows begin {1}, {0,1}, {0,-1,1}, {0,2,-2,1}, {0,-5,5,-3,1}, ...
Triangle begins
  1;
  0,    1;
  0,   -1,    1;
  0,    2,   -2,   1;
  0,   -5,    5,  -3,    1;
  0,   14,  -14,   9,   -4,   1;
  0,  -42,   42, -28,   14,  -5,  1;
  0,  132, -132,  90,  -48,  20, -6,  1;
  0, -429,  429, -297, 165, -75, 27, -7, 1;
Production matrix is
  0,  1,
  0, -1,  1,
  0,  1, -1,  1,
  0, -1,  1, -1,  1,
  0,  1, -1,  1, -1,  1,
  0, -1,  1, -1,  1, -1,  1,
  0,  1, -1,  1, -1,  1, -1,  1,
  0, -1,  1, -1,  1, -1,  1, -1,  1,
  0,  1, -1,  1, -1,  1, -1,  1, -1,  1

Mathematica

T[n_, k_]:= If[n == 0 && k == 0, 1, If[n == 0 && k > 0, 0, (-1)^(n + k)*Binomial[2*n - k - 1, n - k]*k/n]];  Table[T[n, k], {n, 0, 15}, {k, 0, n}] // Flatten (* G. C. Greubel, Dec 31 2017 *)

Prog

(PARI) {T(n, k) = if(n == 0 && k == 0, 1, if(n == 0 && k > 0, 0, (-1)^(n + k)*binomial(2*n - k - 1, n - k)*k/n))};
for(n=0, 15, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Dec 31 2017

Crossrefs

The three triangles A059365, A106566 and A099039 are the same except for signs and the leading term.

Cf. A033184, A000108.

Cf. A106566 (unsigned version), A059365

The following are all versions of (essentially) the same Catalan triangle: A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.

Diagonals give A000108 A000245 A002057 A000344 A003517 A000588 A003518 A003519 A001392, ...

Keyword

easy,sign,tabl

Author

Paul Barry, Sep 23 2004

Status

approved