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A078812 Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1). 43
1, 2, 1, 3, 4, 1, 4, 10, 6, 1, 5, 20, 21, 8, 1, 6, 35, 56, 36, 10, 1, 7, 56, 126, 120, 55, 12, 1, 8, 84, 252, 330, 220, 78, 14, 1, 9, 120, 462, 792, 715, 364, 105, 16, 1, 10, 165, 792, 1716, 2002, 1365, 560, 136, 18, 1, 11, 220, 1287, 3432, 5005, 4368, 2380, 816, 171, 20 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Apart from signs, identical to A053122.

Coefficient array for Morgan-Voyce polynomial B(n,x); see A085478 for references. - Philippe Deléham, Feb 16 2004

T(n,k) = number of compositions of n having k parts when there are q kinds of part q (q=1,2,...). Example: T(4,2) = 10 because we have (1,3),(1,3'),(1,3"), (3,1),(3',1),(3",1),(2,2),(2,2'),(2',2) and (2',2'). - Emeric Deutsch, Apr 09 2005

T(n, k) is also the number of idempotent order-preserving full transformations (of an n-chain) of height k (height(alpha) = |Im(alpha)|). - Abdullahi Umar, Oct 02 2008

A078812 is jointly generated with A085478 as a triangular array of coefficients of polynomials v(n,x): initially, u(1,x) = v(1,x) = 1; for n > 1, u(n,x) = u(n-1,x) + x*v(n-1)x and v(n,x) = u(n-1,x) + (x+1)*v(n-1,x). See the Mathematica section. - Clark Kimberling, Feb 25 2012

Concerning Kimberling's recursion relations, see A102426. - Tom Copeland, Jan 19 2016

Subtriangle of the triangle T(n,k), 0 <= k <= n, read by rows, given by (0, 2, -1/2, 1/2, 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, Mar 27 2012

From Wolfdieter Lang, Aug 30 2012: (Start)

With offset [0,0] the triangle with entries R(n,k) = T(n+1,k+1):= binomial(n+k+1, 2*k+1), n >= k >= 0, and zero otherwise, becomes the Riordan lower triangular convolution matrix R = (G(x)/x, G(x)) with G(x):=x/(1-x)^2 (o.g.f. of A000027). This means that the o.g.f. of column number k of R is (G(x)^(k+1))/x. This matrix R is the inverse of the signed Riordan lower triangular matrix A039598, called in a comment there S.

The Riordan matrix with entries R(n,k), just defined, provides the transition matrix between the sequence entry F(4*m*(n+1))/L(2*l), with m >= 0, for n=0,1,... and the sequence entries 5^k*F(2*m)^(2*k+1) for k = 0,1,...,n, with F=A000045 (Fibonacci) and L=A000032 (Lucas). Proof: from the inverse of the signed triangle Riordan matrix S used in a comment on A039598. (End)

From R. Bagula's comment in A053122 (cf. Damianou link p. 10), this array gives the coefficients (mod sign) of the characteristic polynomials for the Cartan matrix of the root system A_n. - Tom Copeland, Oct 11 2014

For 1 <= k <= n, T(n,k) equals the number of (n-1)-length ternary words containing k-1 letters equal 2 and avoiding 01. - Milan Janjic, Dec 20 2016

REFERENCES

P Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.

LINKS

T. D. Noe, Rows n = 0..50 of triangle, flattened

J. P. Allouche and M. Mendes France, Stern-Brocot polynomials and power series, arXiv preprint arXiv:1202.0211 [math.NT], 2012. - From N. J. A. Sloane, May 10 2012

P. Barry, Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays, JIS 12 (2009) 09.8.6

P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014.

Nour-Eddine Fahssi, On the combinatorics of exclusion in Haldane fractional statistics, arXiv:1808.00045 [cond-mat.stat-mech], 2018.

Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

A. Laradji, and A. Umar, Combinatorial results for semigroups of order-preserving full transformations, Semigroup Forum 72 (2006), 51-62.

Yidong Sun, Numerical triangles and several classical sequences, Fib. Quart. 43, no. 4, (2005) 359-370.

FORMULA

G.f.: x*y/(1-(2+y)*x+x^2). To get row n, expand this in powers of x then expand the coefficient of x^n in increasing powers of y.

If indexing begins at 0 we have T(n, k) = (n+k+1)!/((n-k)!*(2k+1))!. T(n, k) = Sum_{j>=0} T(n-1-j, k-1)*(j+1) with T(n, 0) = n+1, T(n, k) = 0 if n < k. T(n, k) = T(n-1, k-1) + T(n-1, k) + Sum_{j>=0} (-1)^j*T(n-1, k+j)*A000108(j) with T(n, k) = 0 if k<0, T(0, 0)=1 and T(0, k) = 0 for k>0. G.f. for the column k: Sum_{n>=0} T(n, k)*x^n = (x^k)/(1-x)^(2k+2). Row sums: Sum_{k>=0} T(n, k) = A001906(n+1). - Philippe Deléham, Feb 16 2004

Antidiagonal sums are A000079(n) = Sum_{k=0..floor(n/2)} binomial(n+k+1, n-k). - Paul Barry, Jun 21 2004

Riordan array (1/(1-x)^2, x/(1-x)^2). - Paul Barry, Oct 22 2006

T(0,0) = 1, T(n,k) = 0 if k < 0 or if k > n, T(n,k) = T(n-1,k-1) + 2*T(n-1,k) - T(n-2,k). - Philippe Deléham, Jan 26 2010

T(n,m) = Sum_{k=0..n-m}(binomial(2*k,n-m)*binomial(m+k,k)*(-1)^(n-m+k)*binomial(n+1,m+k+1)). - Vladimir Kruchinin, Apr 13 2016

EXAMPLE

Triangle begins, 1 <= k <= n:

                          1

                        2   1

                      3   4   1

                    4  10   6   1

                  5  20  21   8   1

                6  35  56  36  10   1

              7  56 126 120  55  12   1

            8  84 252 330 220  78  14   1

(0, 2, -1/2, 1/2, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, ...) begins:

1

0, 1

0, 2,  1

0, 3,  4,   1

0, 4, 10,   6,   1

0, 5, 20,  21,   8,   1

0, 6, 35,  56,  36,  10,  1

0, 7, 56, 126, 120,  55, 12,  1

0, 8, 84, 252, 330, 220, 78, 14, 1. - Philippe Deléham, Mar 27 2012

For the transition matrix R (T with offset [0,0]) defined in a comment above, row n=2: F(12*m)/L(2*m) = 3*5^0*F(2*m)^1 + 4*5^1*F(2*m)^3 + 1*5^2*F(2*m)^5, m >= 0. - Wolfdieter Lang, Aug 30 2012

MAPLE

for n from 1 to 11 do seq(binomial(n+k-1, 2*k-1), k=1..n) od; # yields sequence in triangular form - Emeric Deutsch

MATHEMATICA

u[1, x_] := 1; v[1, x_] := 1; z = 13;

u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];

v[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];

Table[Expand[u[n, x]], {n, 1, z/2}]

Table[Expand[v[n, x]], {n, 1, z/2}]

cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

TableForm[cu]

Flatten[%]   (* A085478 *)

Table[Expand[v[n, x]], {n, 1, z}]

cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

TableForm[cv]

Flatten[%]   (* A078812 *)

(* Clark Kimberling, Feb 25 2012 *)

PROG

(PARI) {T(n, k) = if( n<0, 0, binomial(n+k-1, 2*k-1))};

(PARI) {T(n, k) = polcoeff( polcoeff( x*y / (1 - (2 + y) * x + x^2) + x * O(x^n), n), k)};

(Haskell)

a078812 n k = a078812_tabl !! n !! k

a078812_row n = a078812_tabl !! n

a078812_tabl = [1] : [2, 1] : f [1] [2, 1] where

   f us vs = ws : f vs ws where

     ws = zipWith (-) (zipWith (+) ([0] ++ vs) (map (* 2) vs ++ [0]))

                      (us ++ [0, 0])

-- Reinhard Zumkeller, Dec 16 2013

(Sage)

@cached_function

def T(k, n):

    if k==n: return 1

    if k==0: return 0

    return sum(i*T(k-1, n-i) for i in (1..n-k+1))

A078812 = lambda n, k: T(k, n)

[[A078812(n, k) for k in (1..n)] for n in (1..8)] # Peter Luschny, Mar 12 2016

(Maxima)

T(n, m):=sum(binomial(2*k, n-m)*binomial(m+k, k)*(-1)^(n-m+k)*binomial(n+1, m+k+1), k, 0, n-m); /* Vladimir Kruchinin, Apr 13 2016 */

(MAGMA) /* As triangle */ [[Binomial(n+k-1, 2*k-1): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Jun 01 2018

CROSSREFS

This triangle is formed from odd-numbered rows of triangle A011973 read in reverse order.

The column sequences are A000027, A000292, A000389, A000580, A000582, A001288 for k=1..6, resp. For k=7..24 they are A010966..(+2)..A011000 and for k=25..50 they are A017713..(+2)..A017763.

Cf. A053123, A049310. Row sums give A001906.

With signs: A053122.

Cf. A119900. - Philippe Deléham, Dec 02 2008

Cf. A085478.

Cf. A029653, A106195.

Sequence in context: A143326 A186686 A053122 * A104711 A133112 A247239

Adjacent sequences:  A078809 A078810 A078811 * A078813 A078814 A078815

KEYWORD

easy,nice,nonn,tabl,changed

AUTHOR

Michael Somos, Dec 05 2002

EXTENSIONS

Edited by N. J. A. Sloane, Apr 28 2008

STATUS

approved

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Last modified November 20 10:52 EST 2018. Contains 317385 sequences. (Running on oeis4.)