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

0,3

Comments

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.

Links

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

Formula

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

Example

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

Maple

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

Mathematica

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 *)

Crossrefs

Cf. A096161, row sums of A333144.

Sequence in context: A162560 A293250 A096161 * A074501 A009123 A066764

Adjacent sequences: A161776 A161777 A161778 * A161780 A161781 A161782

Keyword

nonn

Author

Gary W. Adamson, Jun 19 2009

Extensions

Extended by R. J. Mathar, Jun 23 2009

Status

approved