A215915 - OEIS

%I #20 Oct 18 2017 05:46:02

%S 1,1,3,13,79,579,5209,53347,628257,8223481,119473291,1893056781,

%T 32677209103,606930554923,12109058077809,257638964244739,

%U 5830359141736129,139638723615395697,3531794326401241747,93977250969358226701,2625647922067519041231,76809884197769914248211

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

%C 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.

%H Seiichi Manyama, <a href="/A215915/b215915.txt">Table of n, a(n) for n = 0..432</a>

%F 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

%e 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! +  ...

%e 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 + ...

%t 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 *)

%o (PARI) {a(n)=n!*polcoeff(exp(sum(m=1,n+1,numbpart(m)*x^m/m+x*O(x^n))),n)}

%o for(n=0,31,print1(a(n),", "))

%o (Maxima)

%o 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 */

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

%K nonn

%O 0,3

%A _Paul D. Hanna_, Aug 26 2012