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

0,5

Comments

Square of A106566. Row sums are A127632.

Links

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

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

Example

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;

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 *)

Prog

(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
(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
(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

Crossrefs

Cf. A106566, A121988, A129442.

Sequence in context: A275328 A147720 A205813 * A122538 A090238 A358694

Adjacent sequences: A127628 A127629 A127630 * A127632 A127633 A127634

Keyword

easy,nonn,tabl

Author

Paul Barry, Jan 20 2007

Status

approved