A287532 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals upwards, where A(n,k) = sum of unimodal products of length n and bound k.
1, 1, 1, 1, 4, 1, 1, 11, 9, 1, 1, 26, 50, 16, 1, 1, 57, 222, 150, 25, 1, 1, 120, 867, 1080, 355, 36, 1, 1, 247, 3123, 6627, 3775, 721, 49, 1, 1, 502, 10660, 36552, 33502, 10626, 1316, 64, 1, 1, 1013, 35064, 187000, 262570, 128758, 25676, 2220, 81, 1
Offset: 0
Examples
A(2,3)=50 because of the products 1*1,1*1,1*1 [m=0,1,2]; 1*2,1*2 [m=1,2]; 1*3; 2*1,2*1 [m=0,1]; 2*2,2*2,2*2 [m=0,1,2]; 2*3; 3*1; 3*2; 3*3; total 50. Square array begins: n\k| 1, 2, 3, 4, 5, 6, ... ---+------------------------------------------ 0 | 1, 1, 1, 1, 1, 1, ... 1 | 1, 4, 9, 16, 25, 36, ... 2 | 1, 11, 50, 150, 355, 721, ... 3 | 1, 26, 222, 1080, 3775, 10626, ... 4 | 1, 57, 867, 6627, 33502, 128758, ... 5 | 1, 120, 3123, 36552, 262570, 1360128, ... ...
Crossrefs
Programs
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Mathematica
f[k_]:=Product[1-j x,{j,k}]; A[n_,k_]:=Coefficient[Series[1/f[k]/f[k-1],{x,0,n}],x,n] -
PARI
a(n, k) = sum(j=0, n, stirling(j+k-1, k-1, 2)*stirling(n-j+k, k, 2)); \\ Seiichi Manyama, May 14 2025
Formula
A(n,k) is the coefficient of x^n in 1/((1-k*x) * (1-(k-1)*x)^2 * ... * (1-x)^2).
A(n,k) = Sum_{j=0..n} Stirling2(j+k-1,k-1) * Stirling2(n-j+k,k) for k >= 1. - Seiichi Manyama, May 14 2025
Comments