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