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A077365 Sum of products of factorials of parts in all partitions of n. 17
1, 1, 3, 9, 37, 169, 981, 6429, 49669, 430861, 4208925, 45345165, 536229373, 6884917597, 95473049469, 1420609412637, 22580588347741, 381713065286173, 6837950790434781, 129378941557961565, 2578133190722896861, 53965646957320869469, 1183822028149936497501 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

LINKS

Vincenzo Librandi and 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.

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

EXAMPLE

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 + ...

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}]

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 *)

PROG

(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 */

CROSSREFS

Cf. A006906, A074141, A256125, A265950.

Sequence in context: A002751 A245890 A119856 * A319119 A006229 A321734

Adjacent sequences:  A077362 A077363 A077364 * A077366 A077367 A077368

KEYWORD

nonn

AUTHOR

Vladeta Jovovic, Nov 30 2002

EXTENSIONS

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

New Mathematica code by Olivier Gérard, Oct 22 2011

STATUS

approved

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Last modified April 18 22:08 EDT 2019. Contains 322237 sequences. (Running on oeis4.)