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A127631 Square of Riordan array (1, x*c(x)) where c(x) is the g.f. of A000108. 2
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 (list; table; graph; refs; listen; history; text; internal format)
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

   else: return (k/n)*sum(binomial(2*j+k-1, j)* binomial(2*n-k-j-1, n-k-j) for j in (0..n-k))

[[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 A047922

Adjacent sequences:  A127628 A127629 A127630 * A127632 A127633 A127634

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, Jan 20 2007

STATUS

approved

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Last modified June 2 17:02 EDT 2020. Contains 334787 sequences. (Running on oeis4.)