A293528 E.g.f.: exp(x * Product_{k>0} (1 + x^k)).

1, 1, 3, 13, 97, 741, 7291, 81313, 1027713, 14231017, 220911571, 3730744821, 68096325793, 1339705629133, 28225576881867, 634123159354441, 15127595174135041, 381586517104288593, 10147599723510322723, 283846981316172613597, 8324822922497497733601

Offset: 0

Author

Seiichi Manyama, Oct 11 2017

Keywords

• nonn
• easy

Comments

From Peter Bala, Mar 28 2022: (Start)

The congruence a(n+k) == a(n) (mod k) holds for all n and k.

It follows that the sequence obtained by taking a(n) modulo a fixed positive integer k is periodic with exact period dividing k. For example, the sequence taken modulo 10 becomes [1, 1, 3, 3, 7, 1, 1, 3, 3, 7, ...], a purely periodic sequence with period 5.

3 divides a(3*n+2); 13 divides a(13*n+3) and a(13*n+5); 19 divides a(19*n+5), a(19*n+12) and a(19*n+14). (End)

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..431
• Peter Bala, Integer sequences that become periodic on reduction modulo k for all k

Crossrefs

Cf. A000009, A293527, A293840.

Programs

Mathematica

nmax = 25; CoefficientList[Series[E^(x*QPochhammer[-1, x]/2), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 11 2017 *)

PARI

N=66; x='x+O('x^N); Vec(serlaplace(exp(x*prod(k=1, N, (1+x^k)))))

Formula

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A000009(k-1)*a(n-k)/(n-k)! for n > 0.