A333144 -id:A333144 - cp's OEIS frontend

A161779 The sequence of factorials convolved with all its regularly "aerated" variants.

1, 1, 3, 8, 30, 133, 768, 5221, 41302, 369170, 3677058, 40338310, 483134179, 6271796072, 87709287104, 1314511438945, 21017751750506, 357102350816602, 6424883282375340, 122025874117476166, 2439726373093186274, 51220112287152570828, 1126575412217509969515

Offset: 0

Gary W. Adamson, Jun 19 2009

nonn

Essentially a duplicate of A096161: 1, followed by A096161.

Convolve A000142 = 1,1,2,6,24,... with 1,0,1,0,2,0,6,0,24,.. and with 1,0,0,1,0,0,2,0,0,6,0,0,24,0,0,.. and with 1,0,0,0,1,0,0,0,2,0,0,0,6,... etc.

			Let the partial products = a, a*b, a*b*c,..., with the first few rows =
(1, 1, 2, 6, 24, 120,...) = a
(1, 1, 3, 7, 28, 128,...) = a*b
(1, 1, 3, 8, 29, 131,...) = a*b*c
(1, 1, 3, 8, 30, 132,...) = a*b*c*d
...converging to A161779

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A096161, row sums of A333144.

read("transforms3") ; read("transforms") ; A161779 := proc(N) local a000142,res,n,j ; a000142 := [seq(n!,n=0..N)] ; res := [seq(op(n,a000142),n=1..N)] ; for j from 1 to N do res := CONV( res, AERATE(a000142,j)) ; od: [seq(op(n,res),n=1..N)] end: A161779(30) ; # R. J. Mathar, Jun 23 2009
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0 or i=1, n!,
      add(b(n-i*j, i-1)*j!, j=0..n/i))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 03 2018, revised, Mar 05 2024

b[n_, i_] := b[n, i] = If[i>n, 0, If[Mod[n, i] == 0, (n/i)!, 0] + Sum[j! b[n - i j, i + 1], {j, 0, n/i}]];
a[n_] := If[n == 0, 1, b[n, 1]];
a /@ Range[0, 25] (* Jean-François Alcover, Feb 04 2020, after Alois P. Heinz *)

a(n) = A096161(n) for n >= 1. - R. J. Mathar, Jun 26 2009

a(n) ~ n! * (1 + 1/n^2 + 2/n^3 + 7/n^4 + 28/n^5 + 121/n^6 + 587/n^7 + 3205/n^8 + 19201/n^9 + 123684/n^10), for coefficients see A293266. - Vaclav Kotesovec, Oct 04 2017

Extended by R. J. Mathar, Jun 23 2009

A069123 Triangle formed as follows: For n-th row, n >= 0, record the A000041(n) partitions of n; for each partition, write down number of ways to arrange the parts.

Original entry on oeis.org

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

Offset: 0

Franklin T. Adams-Watters, Apr 07 2002

			This is a function of the individual partitions of an integer. For n = 0 to 5 the terms are (1), (1), (2,1), (6,2,1), (24,6,4,2,1). The partitions are ordered with the largest part sizes first, so the row 4 indices are [4], [3,1], [2,2], [2,1,1] and [1,1,1,1].
.
The irregular table starts:
[0] [1]
[1] [1]
[2] [2, 1]
[3] [6, 2, 1]
[4] [24, 6, 4, 2, 1]
[5] [120, 24, 12, 6, 4, 2, 1]
[6] [720, 120, 48, 24, 36, 12, 6, 8, 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].

Cf. A000142, A333144.

Using Abramowitz-Stegun ordering of partitions this becomes array A134133.

Table[Map[Function[n, Apply[Times, n! ]], IntegerPartitions[i]], {i,0,8}] // Flatten (* Geoffrey Critzer, May 19 2009 *)

def A069123row(n):
    return [product(factorial(part) for part in partition) for partition in Partitions(n)]
for n in (0..6): print(A069123row(n)) # Peter Luschny, Apr 10 2020

[]!=prod_k(n[k]!), or equivalently, []!=prod_k(n[k]!^m[k]).

cp's OEIS Frontend

A161779 The sequence of factorials convolved with all its regularly "aerated" variants.

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