A371567 Array read by downward antidiagonals: A(n,k) = A(n-1,k+1) + (k+1)*Sum_{j=0..k} A(n-1,j) with A(0,k) = k+1, n >= 0, k >= 0.
1, 2, 3, 3, 9, 12, 4, 22, 46, 58, 5, 45, 147, 263, 321, 6, 81, 397, 1012, 1654, 1975, 7, 133, 933, 3341, 7340, 11290, 13265, 8, 204, 1962, 9637, 28333, 56278, 82808, 96073, 9, 297, 3776, 24758, 96313, 246905, 455534, 647680, 743753, 10, 415, 6767, 57678, 292092, 961897, 2227689, 3882510, 5370016, 6113769
Offset: 0
Examples
Array begins: ============================================================== n\k| 0 1 2 3 4 5 6 ... ---+---------------------------------------------------------- 0 | 1 2 3 4 5 6 7 ... 1 | 3 9 22 45 81 133 204 ... 2 | 12 46 147 397 933 1962 3776 ... 3 | 58 263 1012 3341 9637 24758 57678 ... 4 | 321 1654 7340 28333 96313 292092 800991 ... 5 | 1975 11290 56278 246905 961897 3357309 10601156 ... 6 | 13265 82808 455534 2227689 9749034 38415080 137251108 ... ...
Crossrefs
Cf. A258173.
Programs
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PARI
A(m, n=m)={my(r=vectorv(m+1), v=vector(n+m+1, k, k)); r[1] = v[1..n+1]; for(i=1, m, v=vector(#v-1, k, v[k+1] + k*sum(j=1, k, v[j])); r[1+i] = v[1..n+1]); Mat(r)} { A(6) }
Formula
Conjecture: A(n,0) = A258173(n+1). - Mikhail Kurkov, Oct 27 2024
A(n,k) = A(n,k-1) + (A(n,k-1) - A(n-1,k))/k + k*A(n-1,k) + A(n-1,k+1) with A(n,0) = A(n-1,0) + A(n-1,1), A(0,k) = k+1. - Mikhail Kurkov, Nov 24 2024