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A349933 Array read by ascending antidiagonals: the s-th column gives the central s-binomial coefficients.

%I A349933 #15 Feb 13 2024 19:38:20
%S A349933 1,1,1,1,2,1,1,6,3,1,1,20,19,4,1,1,70,141,44,5,1,1,252,1107,580,85,6,
%T A349933 1,1,924,8953,8092,1751,146,7,1,1,3432,73789,116304,38165,4332,231,8,
%U A349933 1,1,12870,616227,1703636,856945,135954,9331,344,9,1,1,48620,5196627,25288120,19611175,4395456,398567,18152,489,10,1
%N A349933 Array read by ascending antidiagonals: the s-th column gives the central s-binomial coefficients.
%H A349933 William Linz, <a href="https://arxiv.org/abs/2110.12095">s-Catalan numbers and Littlewood-Richardson polynomials</a>, arXiv:2110.12095 [math.CO], 2021. See p. 2.
%F A349933 A(n, s) = T(2*n, s*n, s), where T(n, k, s) is the s-binomial coefficient defined as the coefficient of x^k in (Sum_{i=0..s} x^i)^n.
%e A349933 The array begins:
%e A349933 n\s |   0     1     2     3     4
%e A349933 ----+----------------------------
%e A349933   0 |   1     1     1     1     1 ...
%e A349933   1 |   1     2     3     4     5 ...
%e A349933   2 |   1     6    19    44    85 ...
%e A349933   3 |   1    20   141   580  1751 ...
%e A349933   4 |   1    70  1107  8092 38165 ...
%e A349933   ...
%t A349933 T[n_,k_,s_]:=If[k==0,1,Coefficient[(Sum[x^i,{i,0,s}])^n,x^k]]; A[n_,s_]:=T[2n,s n,s]; Flatten[Table[A[n-s,s],{n,0,9},{s,0,n}]]
%Y A349933 Cf. A000984 (s=1), A082758 (s=2), A005721 (s=3), A349936 (s=4), A063419 (s=5), A270918 (n=s), A163269 (s>0).
%Y A349933 Cf. A007318, A008287, A025012, A027907, A035343, A063260.
%Y A349933 Cf. A349934, A349935.
%K A349933 nonn,easy,tabl
%O A349933 0,5
%A A349933 _Stefano Spezia_, Dec 06 2021