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A349935 Array read by ascending antidiagonals: A(n, k) is the n-th spin s-Catalan number, with s = k/2.

%I A349935 #7 Dec 10 2021 11:28:49
%S A349935 1,0,1,2,1,1,0,3,0,1,5,6,4,1,1,0,15,0,5,0,1,14,36,34,16,6,1,1,0,91,0,
%T A349935 65,0,7,0,1,42,232,364,260,111,31,8,1,1,0,603,0,1085,0,175,0,9,0,1,
%U A349935 132,1585,4269,4600,2666,981,260,51,10,1,1,0,4213,0,19845,0,5719,0,369,0,11,0,1
%N A349935 Array read by ascending antidiagonals: A(n, k) is the n-th spin s-Catalan number, with s = k/2.
%H A349935 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. 3.
%F A349935 A(n, k) = T(n, k*(n+1)/2, k) - T(n, k*(n+1)/2+1, k), 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 A349935 A(n, 1) = A126120(n+1).
%F A349935 A(n, 2) = A005043(n+1).
%F A349935 A(3, n) = A000027(n+1).
%F A349935 A(4, 2*n) = A005891(n).
%F A349935 A(5, n) = A006003(n+1).
%e A349935 The array begins:
%e A349935 n\k | 1    2    3    4    5    6
%e A349935 ----+---------------------------
%e A349935   1 | 1    1    1    1    1    1 ...
%e A349935   2 | 0    1    0    1    0    1 ...
%e A349935   3 | 2    3    4    5    6    7 ...
%e A349935   4 | 0    6    0   16    0   31 ...
%e A349935   5 | 5   15   34   65  111  175 ...
%e A349935   6 | 0   36    0  260    0  981 ...
%e A349935   ...
%t A349935 T[n_,k_,s_]:=If[k==0,1,Coefficient[(Sum[x^i,{i,0,s}])^n,x^k]]; A[n_,k_]:=T[n,k(n+1)/2,k]-T[n,k(n+1)/2+1,k]; Flatten[Table[A[n-k+1,k],{n,12},{k,n}]]
%Y A349935 Cf. A000012 (1st row), A059841 (2nd row).
%Y A349935 Cf. A000027, A005043, A005891, A006003, A126120.
%Y A349935 Cf. A349934.
%K A349935 nonn,easy,tabl
%O A349935 1,4
%A A349935 _Stefano Spezia_, Dec 06 2021