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

0,2

Links

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

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).

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 *)

Crossrefs

Cf. A305962, A306026.

Sequence in context: A132745 A124538 A124627 * A064095 A061935 A067078

Adjacent sequences: A305968 A305969 A305970 * A305972 A305973 A305974

Keyword

nonn

Author

Alois P. Heinz, Jun 15 2018

Status

approved