A069123 -id:A069123 - cp's OEIS frontend

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

Vladeta Jovovic, Nov 30 2002

nonn

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

			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 + ...

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.

Cf. A006906, A074141, A256125, A265950.
Cf. A069123, A134133, A134134.
Cf. A051296 (with compositions instead of partitions).

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

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 *)

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

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

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

A134133 A certain partition array in Abramowitz-Stegun order (A-St order).

Original entry on oeis.org

1, 2, 1, 6, 2, 1, 24, 6, 4, 2, 1, 120, 24, 12, 6, 4, 2, 1, 720, 120, 48, 36, 24, 12, 8, 6, 4, 2, 1, 5040, 720, 240, 144, 120, 48, 36, 24, 24, 12, 8, 6, 4, 2, 1, 40320, 5040, 1440, 720, 576, 720, 240, 144, 96, 72, 120, 48, 36, 24, 16, 24, 12, 8, 6, 4, 2, 1, 362880, 40320, 10080

Offset: 1

Wolfdieter Lang, Oct 12 2007

The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].

Partition number array M_3(2)= A130561 divided by partition number array M_3 = M_3(1) = A036040.

			[1], [2,1], [6,2,1], [24,6,4,2,1], [120,24,12,6,4,2,1], ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Wolfdieter Lang, First 10 rows and more.

With another ordering of the partitions this becomes A069123.

Cf. A134134 (triangle obtained by summing same m numbers).

a(n,k) = A130561(n,k)/A036040(n,k) (division of partition arrays M_3(2) by M_3).

a(n,k) = product(j!^e(n,k,j),j=1..n) with the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n.

A333144 Irregular triangle where row n lists the product of the factorials of the exponentials of the partitions of n and the partitions are enumerated in canonical order.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 6, 1, 1, 2, 2, 24, 1, 1, 1, 2, 2, 6, 120, 1, 1, 1, 2, 2, 1, 6, 6, 4, 24, 720, 1, 1, 1, 2, 1, 1, 6, 2, 2, 2, 24, 6, 12, 120, 5040, 1, 1, 1, 2, 1, 1, 6, 2, 1, 2, 2, 24, 2, 4, 2, 6, 120, 24, 12, 48, 720, 40320

Offset: 0

Peter Luschny, Apr 10 2020

By 'canonical order' we understand the graded reverse lexicographic order (the default order of Mathematica and SageMath).

			The irregular table starts:
[0] [1]
[1] [1]
[2] [1, 2]
[3] [1, 1, 6]
[4] [1, 1, 2, 2, 24]
[5] [1, 1, 1, 2, 2, 6, 120]
[6] [1, 1, 1, 2, 2, 1, 6, 6, 4, 24, 720]
[7] [1, 1, 1, 2, 1, 1, 6, 2, 2, 2, 24, 6, 12, 120, 5040]

Row sums are A161779.

Cf. A069123.

def A333144row(n):
    return [product(factorial(expo) for expo in partition.to_exp()) for partition in Partitions(n)]
for n in (0..9): print(A333144row(n))

cp's OEIS Frontend

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

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A134133 A certain partition array in Abramowitz-Stegun order (A-St order).

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A333144 Irregular triangle where row n lists the product of the factorials of the exponentials of the partitions of n and the partitions are enumerated in canonical order.

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Author

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Examples

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Programs

SageMath