A077365 Sum of products of factorials of parts in all partitions of n.

1, 1, 3, 9, 37, 169, 981, 6429, 49669, 430861, 4208925, 45345165, 536229373, 6884917597, 95473049469, 1420609412637, 22580588347741, 381713065286173, 6837950790434781, 129378941557961565, 2578133190722896861, 53965646957320869469, 1183822028149936497501

Offset: 0

Author

Vladeta Jovovic, Nov 30 2002

Keywords

• nonn

Comments

Row sums of arrays A069123 and A134133. Row sums of triangle A134134.

Examples

			The partitions of 4 are 4, 3+1, 2+2, 2+1+1, 1+1+1+1, the corresponding products of factorials of parts are 24,6,4,2,1 and their sum is a(4) = 37.
1 + x + 3 x^2 + 9 x^3 + 37 x^4 + 169 x^5 + 981 x^6 + 6429 x^7 + 49669 x^8 + ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..450 (terms n = 0..70 from Vincenzo Librandi)
• J.-P. Bultel, A, Chouria, J.-G. Luque and O. Mallet, Word symmetric functions and the Redfield-Polya theorem, 2013.

Crossrefs

Cf. A006906, A074141, A256125, A265950.

Cf. A069123, A134133, A134134.

Cf. A051296 (with compositions instead of partitions).

Programs

Maple

b:= proc(n, i, j) option remember;
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, j)+
      `if`(i>n, 0, j^i*b(n-i, i, j+1))))
    end:
a:= n-> b(n$2, 1):
seq(a(n), n=0..40);  # Alois P. Heinz, Aug 03 2013
# second Maple program:
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!)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, May 11 2016

Mathematica

Table[Plus @@ Map[Times @@ (#!) &, IntegerPartitions[n]], {n, 0, 20}] (* Olivier Gérard, Oct 22 2011 *)
a[ n_] := If[ n < 0, 0, Plus @@ Times @@@ (IntegerPartitions[ n] !)] (* Michael Somos, Feb 09 2012 *)
nmax=20; CoefficientList[Series[Product[1/(1-k!*x^k),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Mar 14 2015 *)
b[n_, i_, j_] := b[n, i, j] = If[n==0, 1, If[i<1, 0, b[n, i-1, j] + If[i>n, 0, j^i*b[n-i, i, j+1]]]]; a[n_] := b[n, n, 1]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Oct 12 2015, after Alois P. Heinz *)

PARI

N=66; q='q+O('q^N);
gf= 1/prod(n=1,N, (1-n!*q^n) );
Vec(gf)
/* Joerg Arndt, Oct 06 2012 */

Formula

G.f.: 1/Product_{m>0} (1-m!*x^m).

Recurrence: a(n) = 1/n*Sum_{k=1..n} b(k)*a(n-k), where b(k) = Sum_{d divides k} d*d!^(k/d).

a(n) ~ n! * (1 + 1/n + 3/n^2 + 12/n^3 + 67/n^4 + 457/n^5 + 3734/n^6 + 35741/n^7 + 392875/n^8 + 4886114/n^9 + 67924417/n^10), for coefficients see A256125. - Vaclav Kotesovec, Mar 14 2015

G.f.: exp(Sum_{k>=1} Sum_{j>=1} (j!)^k*x^(j*k)/k). - Ilya Gutkovskiy, Jun 18 2018

Extensions

Unnecessarily complicated mma code deleted by N. J. A. Sloane, Sep 21 2009