A293266 -id:A293266 - cp's OEIS frontend

A096161 Row sums for triangle A096162.

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

Offset: 1

Alford Arnold, Jun 18 2004

nonn

Also, partitions such that a set of k equal terms are labeled 1 through k and can appear in any order. For example, the partition 3+2+2+2+1+1+1+1 of 13 appears 1!*3!*4!=144 times because there are 1! ways to order the one "3," 3! ways to order the three "2"s, ... - Christian G. Bower, Jan 17 2006

			1 1 2 1 3 6 1 4 6 12 24 ... A036038
1 1 1 1 3 1 1 4 3 6 1 ... A036040
1 1 2 1 1 6 1 1 2 2 24 ... A096162
so a(n) begins 1 3 8 30 ... A096161

Seiichi Manyama, Table of n, a(n) for n = 1..449

Cf. A005651, A000110, A036038, A036040, A096162, A110143, A287899.

nmax = 25; Rest[CoefficientList[Series[Product[Sum[k!*x^(j*k), {k, 0, nmax/j}], {j, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 10 2019 *)
m = 25; Rest[CoefficientList[Series[Product[-Gamma[0, -1/x^j] * Exp[-1/x^j], {j, 1, m}] / x^(m*(m + 1)/2), {x, 0, m}], x]] (* Vaclav Kotesovec, Dec 07 2020 *)

{ my(n=25); Vec(prod(k=1, n, O(x*x^n) + sum(r=0, n\k, x^(r*k)*r!))) }

G.f.: B(x)*B(x^2)*B(x^3)*... where B(x) is g.f. of A000142. - Christian G. Bower, Jan 17 2006
G.f.: Product_{k>0} Sum_{r>=0} x^(r*k)*r!. - Andrew Howroyd, Dec 22 2017
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, Aug 10 2019

More terms from Vladeta Jovovic, Jun 22 2004

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

Original entry on oeis.org

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

A326984 Coefficients in asymptotic expansion of sequence A309619.

Original entry on oeis.org

1, 0, 1, 1, 3, 13, 57, 271, 1467, 8905, 58965, 420331, 3212391, 26227477, 227640033, 2090172631, 20222758995, 205524856129, 2188159483341, 24344716477411, 282390978550239, 3408195810080461, 42719427069801369, 555174137978970511, 7469189351830156683

Offset: 0

Vaclav Kotesovec, Aug 10 2019

nonn

			A309619(n) / n! ~ 1 + 1/n^2 + 1/n^3 + 3/n^4 + 13/n^5 + 57/n^6 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..122

Cf. A309619, A293266.

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A096161 Row sums for triangle A096162.

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

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A326984 Coefficients in asymptotic expansion of sequence A309619.

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