A155495 Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).

1, 1, 1, 1, 12, 1, 1, 45, 45, 1, 1, 112, 420, 112, 1, 1, 225, 2100, 2100, 225, 1, 1, 396, 7425, 18480, 7425, 396, 1, 1, 637, 21021, 105105, 105105, 21021, 637, 1, 1, 960, 50960, 448448, 900900, 448448, 50960, 960, 1, 1, 1377, 110160, 1559376, 5513508, 5513508, 1559376, 110160, 1377, 1

Offset

0,5

Comments

T(n,k) equals (-1)^k times the coefficient of x^k in 3F2(-n,-n,-n+1/2;1,1/2;x); see Mathematica code below. - John M. Campbell, Oct 23 2011

Links

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

Formula

T(n, k) = binomial(n, k)*binomial(2*n, 2*k).

Sum_{k=0..n} T(n, k) = A288470(n).

Example

Table starts:
  1;
  1,    1;
  1,   12,      1;
  1,   45,     45,       1;
  1,  112,    420,     112,        1;
  1,  225,   2100,    2100,      225,        1;
  1,  396,   7425,   18480,     7425,      396,        1;
  1,  637,  21021,  105105,   105105,    21021,      637,       1;
  1,  960,  50960,  448448,   900900,   448448,    50960,     960,      1;
  1, 1377, 110160, 1559376,  5513508,  5513508,  1559376,  110160,   1377,    1;
  1, 1900, 218025, 4651200, 26453700, 46558512, 26453700, 4651200, 218025, 1900, 1;

Maple

seq(seq(binomial(2*n, 2*m)*binomial(n, m), m=0..n), n=0..10); # Robert Israel, Jun 12 2017

Mathematica

T[n_, k_]:= Binomial[2*n, 2*k]*Binomial[n, k]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten
Abs[Flatten[Table[CoefficientList[HypergeometricPFQ[{-n, -n, -n+1/2}, {1, 1/2}, x], x], {n, 1, 20}]]] (* or *)
T[n_, k_]:= (-1)^k*Coefficient[HypergeometricPFQ[{-n, -n, -n+1/2}, {1, 1/2}, x], x^k] (* John M. Campbell, Oct 23 2011 *)

Prog

(Magma) [Binomial(n, k)*Binomial(2*n, 2*k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 29 2021
(Sage) flatten([[binomial(n, k)*binomial(2*n, 2*k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 29 2021

Crossrefs

Cf. A155497, A155516, A288470 (row sums).

Sequence in context: A176489 A174039 A174148 * A157273 A350729 A168518

Adjacent sequences: A155492 A155493 A155494 * A155496 A155497 A155498

Keyword

nonn,tabl

Author

Roger L. Bagula, Jan 23 2009

Status

approved