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

Author

Gary W. Adamson, Jun 19 2009

Keywords

• nonn

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.

Examples

			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
		

Links

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

Crossrefs

Cf. A096161, row sums of A333144.

Programs

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 *)

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

Extensions

Extended by R. J. Mathar, Jun 23 2009