A369595 Array read by downward antidiagonals: A(n,k) = A(n-1,k+2) + Sum_{j=0..k} binomial(k,j)*A(n-1,j) with A(0,k) = 1, n >= 0, k >= 0.
1, 1, 2, 1, 3, 7, 1, 5, 14, 37, 1, 9, 30, 89, 264, 1, 17, 68, 227, 737, 2433, 1, 33, 162, 611, 2169, 7696, 27913, 1, 65, 404, 1727, 6695, 25480, 98093, 386906, 1, 129, 1050, 5099, 21573, 87964, 358993, 1490687, 6346119, 1, 257, 2828, 15647, 72287, 315688, 1364681, 5959213, 26542518, 121159373
Offset: 0
Examples
Array begins: ============================================================= n\k| 0 1 2 3 4 5 6 ... ---+--------------------------------------------------------- 0 | 1 1 1 1 1 1 1 ... 1 | 2 3 5 9 17 33 65 ... 2 | 7 14 30 68 162 404 1050 ... 3 | 37 89 227 611 1727 5099 15647 ... 4 | 264 737 2169 6695 21573 72287 251109 ... 5 | 2433 7696 25480 87964 315688 1174756 4522480 ... 6 | 27913 98093 358993 1364681 5376121 21901073 92076673 ... ...
Crossrefs
Cf. A135920.
Programs
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PARI
A(m, n=m)={my(r=vectorv(m+1), v=vector(n+2*m+1, k, 1)); r[1] = v[1..n+1]; for(i=1, m, v=vector(#v-2, k, v[k+2] + sum(j=1, k, binomial(k-1, j-1)*v[j])); r[1+i] = v[1..n+1]); Mat(r)} { A(6) }
Formula
Conjecture: A(n,0) = A135920(n+1). - Mikhail Kurkov, Oct 27 2024