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

Author

Alford Arnold, May 12 2005

Keywords

• easy
• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

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

Cf. A077365.

Programs

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 *)

Maxima

S(n,m):=if n=0 then 1 else if nVladimir Kruchinin, Sep 07 2014 */

Formula

For partition [], 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

Extensions

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

More terms from Vladeta Jovovic, Jul 10 2007