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

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

Offset

0,3

Comments

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.

Links

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

Formula

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

Example

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 + ...

Mathematica

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 *)

Prog

(PARI) {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), ", "))
(Maxima)
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 */

Crossrefs

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

Sequence in context: A141762 A062872 A288954 * A159312 A355165 A213527

Adjacent sequences: A215912 A215913 A215914 * A215916 A215917 A215918

Keyword

nonn

Author

Paul D. Hanna, Aug 26 2012

Status

approved