A293841 -id:A293841 - cp's OEIS frontend

A293840 E.g.f.: exp(Sum_{n>=1} A000009(n)*x^n).

1, 1, 3, 19, 121, 1041, 10651, 121843, 1575729, 22970881, 366805171, 6365365491, 120044573353, 2430782532049, 52677233993931, 1217023986185491, 29799465317716321, 771272544315151233, 21044341084622337379, 603173026772647474771

Offset: 0

Seiichi Manyama, Oct 17 2017

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, 9, 1, 1, 1, 3, 9, 1, ...], a purely periodic sequence with exact period 5.
3 divides a(3*n+2); 9 divides a(9*n+8); 11 divides a(11*n+4); 19 divides a(19*n+3). (End)

Seiichi Manyama, Table of n, a(n) for n = 0..430
Peter Bala, Integer sequences that become periodic on reduction modulo k for all k

Cf. A293839, A293841.

Cf. A058892.

nmax = 20; CoefficientList[Series[E^Sum[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*A000009(k)*a(n-k)/(n-k)! for n > 0.

A293839 E.g.f.: exp(Sum_{n>=1} A000009(n)*x^n/n).

Original entry on oeis.org

1, 1, 2, 8, 38, 238, 1828, 16096, 160604, 1826684, 23018264, 316422304, 4755059848, 77084268712, 1343682876272, 25097562397952, 498130253334032, 10479084018025744, 233353674153699616, 5470193826634531456, 134766983204541259616, 3482705318091355591136

Offset: 0

Seiichi Manyama, Oct 17 2017

nonn

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

Cf. A293840, A293841.

Cf. A215915.

nmax = 20; CoefficientList[Series[E^Sum[PartitionsQ[k]*x^k/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} A000009(k)*a(n-k)/(n-k)! for n > 0.

A293908 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j>=1} j^(k-1)A000009(j)x^j).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 8, 1, 1, 5, 19, 38, 1, 1, 9, 49, 121, 238, 1, 1, 17, 133, 409, 1041, 1828, 1, 1, 33, 373, 1441, 4841, 10651, 16096, 1, 1, 65, 1069, 5233, 23601, 66541, 121843, 160604, 1, 1, 129, 3109, 19441, 119441, 442681, 1006825, 1575729, 1826684, 1, 1

Offset: 0

Seiichi Manyama, Oct 19 2017

			Square array begins:
     1,    1,    1,     1,      1, ...
     1,    1,    1,     1,      1, ...
     2,    3,    5,     9,     17, ...
     8,   19,   49,   133,    373, ...
    38,  121,  409,  1411,   5233, ...
   238, 1041, 4841, 23601, 119441, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..2 give A293839, A293840, A293841.
Rows n=0-1 give A000012.
Cf. A293796.

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

cp's OEIS Frontend

A293840 E.g.f.: exp(Sum_{n>=1} A000009(n)*x^n).

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A293839 E.g.f.: exp(Sum_{n>=1} A000009(n)*x^n/n).

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A293908 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j>=1} j^(k-1)A000009(j)x^j).

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cp's OEIS Frontend

A293840 E.g.f.: exp(Sum_{n>=1} A000009(n)*x^n).

Views

Author

Keywords

Comments

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Crossrefs

Programs

Mathematica

Formula

A293839 E.g.f.: exp(Sum_{n>=1} A000009(n)*x^n/n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A293908 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j>=1} j^(k-1)*A000009(j)*x^j).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A293908 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j>=1} j^(k-1)A000009(j)x^j).