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

Author

Paul D. Hanna, Aug 26 2012

Keywords

• nonn

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.

Examples

			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 + ...
		

Links

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

Crossrefs

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

Programs

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 *)

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 */

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),", "))

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