A349933 Array read by ascending antidiagonals: the s-th column gives the central s-binomial coefficients.
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, 1, 1, 924, 8953, 8092, 1751, 146, 7, 1, 1, 3432, 73789, 116304, 38165, 4332, 231, 8, 1, 1, 12870, 616227, 1703636, 856945, 135954, 9331, 344, 9, 1, 1, 48620, 5196627, 25288120, 19611175, 4395456, 398567, 18152, 489, 10, 1
Offset: 0
Examples
The array begins: n\s | 0 1 2 3 4 ----+---------------------------- 0 | 1 1 1 1 1 ... 1 | 1 2 3 4 5 ... 2 | 1 6 19 44 85 ... 3 | 1 20 141 580 1751 ... 4 | 1 70 1107 8092 38165 ... ...
Links
- William Linz, s-Catalan numbers and Littlewood-Richardson polynomials, arXiv:2110.12095 [math.CO], 2021. See p. 2.
Crossrefs
Programs
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Mathematica
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}]]
Formula
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.