A293839 -id:A293839 - 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.

A215915 E.g.f.: exp( Sum_{n>=1} A000041(n)*x^n/n ), where A000041(n) is the number of partitions of n.

Original entry on oeis.org

1, 1, 3, 13, 79, 579, 5209, 53347, 628257, 8223481, 119473291, 1893056781, 32677209103, 606930554923, 12109058077809, 257638964244739, 5830359141736129, 139638723615395697, 3531794326401241747, 93977250969358226701, 2625647922067519041231, 76809884197769914248211

Offset: 0

Paul D. Hanna, Aug 26 2012

nonn

Note that exp( Sum_{k>=1} A183610(n,k)*x^k/k ) is an integer series for row n>=1; the partition numbers, which forms row 0 of table A183610, is the exception.

			G.f.: A(x) = 1 + x + 3*x^2/2! + 13*x^3/3! + 79*x^4/4! + 579*x^5/5! + 5209*x^6/6! +  ...
such that log(A(x)) = x + 2*x^2/2 + 3*x^3/3 + 5*x^4/4 + 7*x^5/5 + 11*x^6/6 + 15*x^7/7 + 22*x^8/8 + ... + A000041(n)*x^n/n + ...

Seiichi Manyama, Table of n, a(n) for n = 0..432

Cf. A183610, A000041, A058892, A293731, A293839.

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

a(n):=if n=0 then 1 else (n-1)!*sum(num_partitions(i+1)*a(n-i-1)/(n-i-1)!,i,0,n-1); /* Vladimir Kruchinin, Feb 27 2015 */

{a(n)=n!*polcoeff(exp(sum(m=1,n+1,numbpart(m)*x^m/m+x*O(x^n))),n)}
for(n=0,31,print1(a(n),", "))

a(n) = (n-1)!*sum(p(i+1)*a(n-i-1)/(n-i-1)!,i,0,n-1), a(0)=1, where p(i) is the number of partitions of n. - Vladimir Kruchinin, Feb 27 2015

A300514 Expansion of e.g.f. exp(Sum_{k>=1} q(k)*x^k/k!), where q(k) = number of partitions of k into distinct parts (A000009).

Original entry on oeis.org

1, 1, 2, 6, 20, 79, 358, 1791, 9854, 58958, 379716, 2617320, 19197327, 149099827, 1221390172, 10515829901, 94865603724, 894302028718, 8788782784778, 89848652800152, 953666248076772, 10491219933196228, 119429574273909421, 1404835599743325765, 17052591331677804136

Offset: 0

Ilya Gutkovskiy, Mar 07 2018

nonn

Exponential transform of A000009.

			E.g.f.: A(x) = 1 + x/1! + 2*x^2/2! + 6*x^3/3! + 20*x^4/4! + 79*x^5/5! + 358*x^6/6! + 1791*x^7/7! + ...

Alois P. Heinz, Table of n, a(n) for n = 0..550
N. J. A. Sloane, Transforms
Index entries for related partition-counting sequences

Cf. A000009, A293839, A293840, A300511, A300515.

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*b(j), j=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 07 2018

nmax = 24; CoefficientList[Series[Exp[Sum[PartitionsQ[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[PartitionsQ[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 24}]

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

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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A215915 E.g.f.: exp( Sum_{n>=1} A000041(n)*x^n/n ), where A000041(n) is the number of partitions of n.

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A300514 Expansion of e.g.f. exp(Sum_{k>=1} q(k)*x^k/k!), where q(k) = number of partitions of k into distinct parts (A000009).

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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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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).