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A349934 Array read by ascending antidiagonals: A(n, s) is the n-th s-Catalan number.

%I A349934 #15 Dec 10 2021 11:28:38
%S A349934 1,2,1,5,3,1,14,15,4,1,42,91,34,5,1,132,603,364,65,6,1,429,4213,4269,
%T A349934 1085,111,7,1,1430,30537,52844,19845,2666,175,8,1,4862,227475,679172,
%U A349934 383251,70146,5719,260,9,1,16796,1730787,8976188,7687615,1949156,204687,11096,369,10,1
%N A349934 Array read by ascending antidiagonals: A(n, s) is the n-th s-Catalan number.
%H A349934 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 A349934 A(n, s) = T(2*n, s*n, s) - T(2*n, s*n+1, 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.
%F A349934 A(2, n) = A000027(n+1).
%F A349934 A(3, n) = A006003(n+1).
%e A349934 The array begins:
%e A349934 n\s |  1    2     3      4      5
%e A349934 ----+----------------------------
%e A349934   1 |  1    1     1      1      1 ...
%e A349934   2 |  2    3     4      5      6 ...
%e A349934   3 |  5   15    34     65    111 ...
%e A349934   4 | 14   91   364   1085   2666 ...
%e A349934   5 | 42  603  4269  19845  70146 ...
%e A349934   ...
%t A349934 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]-T[2n,s n+1,s]; Flatten[Table[A[n-s+1,s],{n,10},{s,n}]]
%o A349934 (PARI) T(n, k, s) = polcoef((sum(i=0, s, x^i))^n, k);
%o A349934 A(n, s) = T(2*n, s*n, s) - T(2*n, s*n+1, s); \\ _Michel Marcus_, Dec 10 2021
%Y A349934 Cf. A000012 (n=1), A220892 (n=4).
%Y A349934 Cf. A000108 (s=1), A099251 (s=2), A264607 (s=3).
%Y A349934 Cf. A000027, A007318, A008287, A027907, A035343, A063260.
%Y A349934 Cf. A349933.
%K A349934 nonn,easy,tabl
%O A349934 1,2
%A A349934 _Stefano Spezia_, Dec 06 2021