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%I #25 Mar 05 2024 09:33:21
%S 1,1,3,8,30,133,768,5221,41302,369170,3677058,40338310,483134179,
%T 6271796072,87709287104,1314511438945,21017751750506,357102350816602,
%U 6424883282375340,122025874117476166,2439726373093186274,51220112287152570828,1126575412217509969515
%N The sequence of factorials convolved with all its regularly "aerated" variants.
%C Essentially a duplicate of A096161: 1, followed by A096161.
%C 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.
%H Seiichi Manyama, <a href="/A161779/b161779.txt">Table of n, a(n) for n = 0..449</a>
%F a(n) = A096161(n) for n >= 1. - _R. J. Mathar_, Jun 26 2009
%F 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
%e Let the partial products = a, a*b, a*b*c,..., with the first few rows =
%e (1, 1, 2, 6, 24, 120,...) = a
%e (1, 1, 3, 7, 28, 128,...) = a*b
%e (1, 1, 3, 8, 29, 131,...) = a*b*c
%e (1, 1, 3, 8, 30, 132,...) = a*b*c*d
%e ...converging to A161779
%p 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
%p # second Maple program:
%p b:= proc(n, i) option remember; `if`(n=0 or i=1, n!,
%p add(b(n-i*j, i-1)*j!, j=0..n/i))
%p end:
%p a:= n-> b(n$2):
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Oct 03 2018, revised, Mar 05 2024
%t 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}]];
%t a[n_] := If[n == 0, 1, b[n, 1]];
%t a /@ Range[0, 25] (* _Jean-François Alcover_, Feb 04 2020, after _Alois P. Heinz_ *)
%Y Cf. A096161, row sums of A333144.
%K nonn
%O 0,3
%A _Gary W. Adamson_, Jun 19 2009
%E Extended by _R. J. Mathar_, Jun 23 2009
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