A316073 -id:A316073 - cp's OEIS frontend

A306173 a(n) is the n-th term of the inverse Euler transform of j-> n^(j-1).

1, 1, 6, 42, 420, 5155, 77658, 1376340, 28133616, 651317463, 16846515510, 481472570920, 15067838554860, 512473599799551, 18821719654854998, 742395982266536520, 31299550394528466960, 1404629090174946183156, 66851805805525048040334, 3363381327122496643643628

Offset: 1

Alois P. Heinz, Jun 23 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..200

Main diagonal of A065177.

Cf. A075147, A316073.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(g(i, k)+j-1, j)*b(n-i*j, i-1,k), j=0..n/i)))
    end:
g:= proc(n, k) option remember; k^(n-1)-b(n, n-1, k) end:
a:= n-> g(n$2):
seq(a(n), n=1..21);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[g[i, k] + j - 1, j]*b[n - i*j, i - 1, k], {j, 0, n/i}]]];
g[n_, k_] := g[n, k] = k^(n - 1) - b[n, n - 1, k];
a[n_] := g[n, n];
a /@ Range[21] (* Jean-François Alcover, Jan 06 2020, after Alois P. Heinz *)

a(n) ~ (1 - exp(-1)) * n^(n-1). - Vaclav Kotesovec, Oct 08 2019

A383033 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) is the n-th term of the inverse Weigh transform of j-> k^(j-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 1, 3, 2, 1, 1, 4, 6, 5, 0, 1, 5, 12, 18, 6, 0, 1, 6, 20, 46, 42, 11, 0, 1, 7, 30, 95, 156, 113, 18, 1, 1, 8, 42, 171, 420, 566, 294, 35, 0, 1, 9, 56, 280, 930, 1930, 2028, 798, 56, 0, 1, 10, 72, 428, 1806, 5185, 8820, 7396, 2128, 105, 0

Offset: 1

Seiichi Manyama, Apr 13 2025

			Square array begins:
  1,  1,   1,    1,    1,     1,     1, ...
  1,  2,   3,    4,    5,     6,     7, ...
  0,  2,   6,   12,   20,    30,    42, ...
  1,  5,  18,   46,   95,   171,   280, ...
  0,  6,  42,  156,  420,   930,  1806, ...
  0, 11, 113,  566, 1930,  5185, 11851, ...
  0, 18, 294, 2028, 8820, 28830, 77658, ...

Christian G. Bower, PARI programs for transforms, 2007.
N. J. A. Sloane, Maple programs for transforms, 2001-2020.

Columns k=1..3 give A209229, A383034, A383035.
Main diagonal gives A316073.
Cf. A383023.

A(n,k) = (1/n) * (k^n - (k-1)^n + Sum_{d

A(n,k) = A383023(n,k) - A383023(n,k-1).

Product_{n>=1} (1 + x^n)^A(n,k) = (1 - (k-1)*x)/(1 - k*x).

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A306173 a(n) is the n-th term of the inverse Euler transform of j-> n^(j-1).

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