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

%I A127631 #27 Sep 08 2022 08:45:29
%S A127631 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 A127631 0,1348,1216,672,260,70,12,1,0,5814,5385,3150,1344,425,96,14,1,0,
%U A127631 25674,24244,14799,6784,2400,646,126,16,1
%N A127631 Square of Riordan array (1, x*c(x)) where c(x) is the g.f. of A000108.
%C A127631 Square of A106566. Row sums are A127632.
%H A127631 G. C. Greubel, <a href="/A127631/b127631.txt">Rows n = 0..100 of triangle, flattened</a>
%F A127631 Riordan array (1, x*c(x)*c(x*c(x))), where c(x) is the g.f. of A000108.
%F A127631 T(n+1,1) = A129442(n) = A121988(n+1). - _Philippe Deléham_, Feb 27 2013
%F A127631 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 A127631 Triangle begins
%e A127631   1;
%e A127631   0,      1;
%e A127631   0,      2,      1;
%e A127631   0,      6,      4,     1;
%e A127631   0,     21,     16,     6,     1;
%e A127631   0,     80,     66,    30,     8,     1;
%e A127631   0,    322,    280,   143,    48,    10,    1;
%e A127631   0,   1348,   1216,   672,   260,    70,   12,   1;
%e A127631   0,   5814,   5385,  3150,  1344,   425,   96,  14,   1;
%e A127631   0,  25674,  24244, 14799,  6784,  2400,  646, 126,  16,  1;
%e A127631   0, 115566, 110704, 69828, 33814, 13002, 3960, 931, 160, 18, 1;
%t A127631 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 A127631 (Maxima)
%o A127631 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 A127631 (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 A127631 (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 A127631 (Sage)
%o A127631 def T(n, k):
%o A127631    if k == n: return 1
%o A127631    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 A127631 [[T(n,k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Apr 05 2019
%Y A127631 Cf. A106566, A121988, A129442.
%K A127631 easy,nonn,tabl
%O A127631 0,5
%A A127631 _Paul Barry_, Jan 20 2007