A293528 -id:A293528 - cp's OEIS frontend

A293841 E.g.f.: exp(Sum_{n>=1} nA000009(n)x^n).

1, 1, 5, 49, 409, 4841, 66541, 1006825, 17349809, 333948529, 6997459861, 159199648961, 3918175462345, 103227624161689, 2901807752857469, 86684932131301561, 2738566218754961761, 91236821580866560865, 3196113263245038385189

Offset: 0

Seiichi Manyama, Oct 17 2017

nonn

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, 5, 9, 9, 1, 1, 5, 9, 9, ...], a purely periodic sequence with exact period 5.
5 divides a(5*n+2), 7 divides a(7*n+3); 17 divides a(17*n+7), a(17*n+8) and a(17*n+11). (End)

Seiichi Manyama, Table of n, a(n) for n = 0..418
Peter Bala, Integer sequences that become periodic on reduction modulo k for all k

Cf. A000009, A293528, A293840.

nmax = 20; CoefficientList[Series[E^Sum[k*PartitionsQ[k]*x^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 18 2017 *)

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k^2*A000009(k)*a(n-k)/(n-k)! for n > 0.

A293527 E.g.f.: exp(x/Product_{k>0} (1 - x^k)).

Original entry on oeis.org

1, 1, 3, 19, 145, 1401, 15331, 198283, 2840769, 45744625, 807769891, 15590922051, 325339538833, 7316871562729, 175934564213955, 4508362093795771, 122558873094082561, 3522465207528093153, 106681726559176156099, 3395601487535927589235, 113287948824653903674641

Offset: 0

Seiichi Manyama, Oct 11 2017

nonn

From Peter Bala, Mar 25 2022: (Start)
The sequence terms are odd. 3 divides a(3*n+2), 5 divides a(5*n+4), 9 divides a(9*n+8), 15 divides a(15*n+14) and 19 divides a(19*n+3).
More generally, 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, taken modulo 7 the sequence becomes [1, 1, 3, 5, 5, 1, 1, 1, 3, 3, 5, 5, 1, 1, ...], a purely periodic sequence with period 7. (End)

Seiichi Manyama, Table of n, a(n) for n = 0..423
Peter Bala, Integer sequences that become periodic on reduction modulo k for all k

Cf. A000041, A293528.

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

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

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A000041(k-1)*a(n-k)/(n-k)! for n > 0.

A293906 E.g.f.: exp(Sum_{n>=1} A000009(n-1)*x^n/n).

Original entry on oeis.org

1, 1, 2, 6, 30, 174, 1284, 11148, 110604, 1215180, 15066936, 205333416, 3031876872, 48536481096, 834192650160, 15357616617744, 301627633467024, 6274718094035088, 137885714638938144, 3199063553591292000, 77943291010448903136, 1991687512094232297696

Offset: 0

Seiichi Manyama, Oct 19 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..439

Cf. A000009, A293528.

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

cp's OEIS Frontend

A293841 E.g.f.: exp(Sum_{n>=1} nA000009(n)x^n).

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A293527 E.g.f.: exp(x/Product_{k>0} (1 - x^k)).

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A293906 E.g.f.: exp(Sum_{n>=1} A000009(n-1)*x^n/n).

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cp's OEIS Frontend

A293841 E.g.f.: exp(Sum_{n>=1} n*A000009(n)*x^n).

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Formula

A293527 E.g.f.: exp(x/Product_{k>0} (1 - x^k)).

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Author

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Programs

Mathematica

PARI

Formula

A293906 E.g.f.: exp(Sum_{n>=1} A000009(n-1)*x^n/n).

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Author

Keywords

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Formula

A293841 E.g.f.: exp(Sum_{n>=1} nA000009(n)x^n).