A107107 For each partition of n, calculate (dM2/dM3) where dM2 = A036039(p) and dM3 = A036040(p); then sum over all partitions of n.

1, 1, 2, 4, 11, 37, 168, 926, 6181, 47651, 418546, 4106264, 44537519, 528408261, 6807428748, 94588717554, 1409927483625, 22437711255279, 379674820846534, 6806486383431340, 128862216628864163, 2569080120361323721, 53797824318887051264, 1180533584545138213222

Offset

0,3

Comments

Values for individual partitions (A107106) are factorials when all but one part of the partition has size one or two, but not usually in other cases.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..448

Formula

For partition [<c_i^k_i>], the contribution to the sum is product_i (c_i - 1)!^k_i.

G.f.: 1/Product_{m>0} (1-(m-1)!*x^m). - Vladeta Jovovic, Jul 10 2007

a(n) = S(n,1), where S(n,m) = sum(k=m..n/2, (k-1)!*S(n-k,k))+(n-1)!, S(n,n)=(n-1)!, S(0,m)=1, S(n,m)=0 for m>n. - Vladimir Kruchinin, Sep 07 2014

a(n) ~ (n-1)! * (1 + 1/n + 3/n^2 + 11/n^3 + 50/n^4 + 278/n^5 + 1861/n^6 + 14815/n^7 + 138477/n^8 + 1497775/n^9 + 18465330/n^10). - Vaclav Kotesovec, Mar 15 2015

Example

For n = 6, (120,144,90,40,90,120,15,40,45,15,1) / (1,6,15,10,15,60,15,20,45,15,1)
  equals (120,24,6,4,6,2,1,2,1,1,1) so A107107(6) = 168.

Maple

b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+
      `if`(i>n, 0, b(n-i, i)*(i-1)!)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, May 11 2016

Mathematica

nmax=20; CoefficientList[Series[Product[1/(1-(k-1)!*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 15 2015 *)

Prog

(Maxima)
S(n, m):=if n=0 then 1 else if n<m then 0 else if n=m then (n-1)! else sum((k-1)!*S(n-k, k), k, m, n/2)+(n-1)!;
makelist(S(n, 1), n, 1, 17); /* Vladimir Kruchinin, Sep 07 2014 */

Crossrefs

Cf. A000142, A036039, A000110, A036040, A107106, A102189.

Cf. A077365.

Sequence in context: A328433 A118182 A179327 * A243565 A101898 A193188

Adjacent sequences: A107104 A107105 A107106 * A107108 A107109 A107110

Keyword

easy,nonn

Author

Alford Arnold, May 12 2005

Extensions

Edited, corrected and extended by Franklin T. Adams-Watters, Nov 03 2005

More terms from Vladeta Jovovic, Jul 10 2007

Status

approved