A305971 Antidiagonal sums of A305962.

1, 2, 3, 5, 11, 34, 141, 736, 4653, 34842, 303848, 3041514, 34520903, 439820187, 6238591638, 97832195694, 1685800545944, 31746373299029, 650170193047230, 14418116545259245, 344857160229381442, 8865220175506008295, 244158955254595904415, 7183277314615065192163

Offset: 0

Author

Alois P. Heinz, Jun 15 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A305962, A306026.

Programs

Maple

b:= proc(n, k, m) option remember; `if`(n=0, 1,
      add(b(n-1, k, max(m, j)), j=1..m+k))
    end:
a:= n-> add(b(j, n-j, 1+j-n), j=0..n):
seq(a(n), n=0..25);
# second Maple program:
b:= (n, k)-> `if`(n=0, 1, (n-1)!*coeff(series(exp(x+add(
              (exp(j*x)-1)/j, j=1..k)), x, n), x, n-1)):
a:= n-> add(b(j, n-j), j=0..n):
seq(a(n), n=0..25);

Mathematica

b[n_, k_, m_] := b[n, k, m] = If[n == 0, 1, Sum[b[n - 1, k, Max[m, j]], {j, 1, m + k}]];
a[n_] := Sum[b[j, n - j, 1 + j - n], {j, 0, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Nov 16 2022, after Alois P. Heinz *)

Formula

a(n) = Sum_{j=0..n} (j-1)! * [x^(j-1)] exp(x + Sum_{i=1..n-j} (exp(i*x)-1)/i) for n > 0, a(0) = 1.

a(n) = Sum_{j=0..n} A305962(j,n-j).