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Square of Riordan array (1, x*c(x)) where c(x) is the g.f. of A000108.
2

%I #27 Sep 08 2022 08:45:29

%S 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,

%T 0,1348,1216,672,260,70,12,1,0,5814,5385,3150,1344,425,96,14,1,0,

%U 25674,24244,14799,6784,2400,646,126,16,1

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

%C Square of A106566. Row sums are A127632.

%H G. C. Greubel, <a href="/A127631/b127631.txt">Rows n = 0..100 of triangle, flattened</a>

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

%F T(n+1,1) = A129442(n) = A121988(n+1). - _Philippe Deléham_, Feb 27 2013

%F 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

%e Triangle begins

%e 1;

%e 0, 1;

%e 0, 2, 1;

%e 0, 6, 4, 1;

%e 0, 21, 16, 6, 1;

%e 0, 80, 66, 30, 8, 1;

%e 0, 322, 280, 143, 48, 10, 1;

%e 0, 1348, 1216, 672, 260, 70, 12, 1;

%e 0, 5814, 5385, 3150, 1344, 425, 96, 14, 1;

%e 0, 25674, 24244, 14799, 6784, 2400, 646, 126, 16, 1;

%e 0, 115566, 110704, 69828, 33814, 13002, 3960, 931, 160, 18, 1;

%t 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 *)

%o (Maxima)

%o 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 */

%o (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

%o (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

%o (Sage)

%o def T(n, k):

%o if k == n: return 1

%o 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

%o [[T(n,k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Apr 05 2019

%Y Cf. A106566, A121988, A129442.

%K easy,nonn,tabl

%O 0,5

%A _Paul Barry_, Jan 20 2007