A127631 Square of Riordan array (1, x*c(x)) where c(x) is the g.f. of A000108.

1, 0, 1, 0, 2, 1, 0, 6, 4, 1, 0, 21, 16, 6, 1, 0, 80, 66, 30, 8, 1, 0, 322, 280, 143, 48, 10, 1, 0, 1348, 1216, 672, 260, 70, 12, 1, 0, 5814, 5385, 3150, 1344, 425, 96, 14, 1, 0, 25674, 24244, 14799, 6784, 2400, 646, 126, 16, 1

Offset: 0

Author

Paul Barry, Jan 20 2007

Keywords

• easy
• nonn
• tabl

Comments

Square of A106566. Row sums are A127632.

Examples

			Triangle begins
  1;
  0,      1;
  0,      2,      1;
  0,      6,      4,     1;
  0,     21,     16,     6,     1;
  0,     80,     66,    30,     8,     1;
  0,    322,    280,   143,    48,    10,    1;
  0,   1348,   1216,   672,   260,    70,   12,   1;
  0,   5814,   5385,  3150,  1344,   425,   96,  14,   1;
  0,  25674,  24244, 14799,  6784,  2400,  646, 126,  16,  1;
  0, 115566, 110704, 69828, 33814, 13002, 3960, 931, 160, 18, 1;
		

Links

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

Crossrefs

Cf. A106566, A121988, A129442.

Programs

Magma

[[k eq n select 1 else (k/n)*(&+[Binomial(2*j+k-1,j)*Binomial(2*n -k-j-1, n-k-j): j in [0..n-k]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Apr 05 2019

Mathematica

T[n_, k_]:= If[k==n, 1, (k/n)*Sum[Binomial[2*j-k-1, j-k]*Binomial[2*n-j- 1, n-j], {j,k,n}]]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 05 2019 *)

Maxima

T(n,k):=if k=n then 1 else if n=0 then 0 else (k*sum((binomial(-k+2*i-1,i-k))*(binomial(2*n-i-1,n-i)),i,k,n))/n; /* Vladimir Kruchinin, Apr 05 2019 */

PARI

{T(n,k) = if(k==n, 1, (k/n)*sum(j=0,n-k, binomial(2*j+k-1, j)* binomial(2*n-k-j-1, n-k-j)))}; \\ G. C. Greubel, Apr 05 2019

Sage

def T(n, k):
   if k == n: return 1
   return (k*sum(binomial(2*j+k-1, j)* binomial(2*n-k-j-1, n-k-j) for j in (0..n-k)))//n
[[T(n,k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Apr 05 2019

Formula

Riordan array (1, x*c(x)*c(x*c(x))), where c(x) is the g.f. of A000108.

T(n+1,1) = A129442(n) = A121988(n+1). - Philippe Deléham, Feb 27 2013

T(n,k) = (k/n)*Sum_{i=k..n} C(2*i-k-1,i-k)*C(2*n-i-1,n-i), T(n,n)=1. - Vladimir Kruchinin, Apr 05 2019