A370380 Array read by downward antidiagonals: A(n,k) = (k+2)*A(n-1,k+1) + Sum_{j=0..k} A(n-1,j) with A(0,k) = 1, n >= 0, k >= 0.
1, 1, 3, 1, 5, 13, 1, 7, 29, 71, 1, 9, 51, 195, 461, 1, 11, 79, 409, 1493, 3447, 1, 13, 113, 737, 3623, 12823, 29093, 1, 15, 153, 1203, 7427, 35285, 122125, 273343, 1, 17, 199, 1831, 13601, 81009, 375591, 1277991, 2829325, 1, 19, 251, 2645, 22961, 164371, 954419, 4344485, 14584789, 31998903
Offset: 0
Examples
Array begins: =========================================================== n\k| 0 1 2 3 4 5 6 ... ---+------------------------------------------------------- 0 | 1 1 1 1 1 1 1 ... 1 | 3 5 7 9 11 13 15 ... 2 | 13 29 51 79 113 153 199 ... 3 | 71 195 409 737 1203 1831 2645 ... 4 | 461 1493 3623 7427 13601 22961 36443 ... 5 | 3447 12823 35285 81009 164371 304667 526833 ... 6 | 29093 122125 375591 954419 2124937 4289433 8025755 ... ...
Crossrefs
Row 2 appears to be essentially A144391. - Joerg Arndt, Feb 17 2024
Cf. A003319.
Programs
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PARI
A(m, n=m)={my(r=vectorv(m+1), v=vector(n+m+1, k, 1)); r[1] = v[1..n+1]; for(i=1, m, v=vector(#v-1, k, (k+1)*v[k+1] + sum(j=1, k, v[j])); r[1+i] = v[1..n+1]); Mat(r)} { A(6) }
Formula
Conjecture: A(n,0) = A003319(n+2). - Mikhail Kurkov, Oct 27 2024
A(n,k) = A(n,k-1) - k*A(n-1,k) + (k+2)*A(n-1,k+1) with A(n,0) = A(n-1,0) + 2*A(n-1,1), A(0,k) = 1. - Mikhail Kurkov, Nov 23 2024