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

1,2

Comments

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

Links

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

Formula

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

Example

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

Mathematica

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 *)

Prog

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

Crossrefs

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

Sequence in context: A348662 A162560 A293250 * A161779 A074501 A009123

Adjacent sequences: A096158 A096159 A096160 * A096162 A096163 A096164

Keyword

nonn

Author

Alford Arnold, Jun 18 2004

Extensions

More terms from Vladeta Jovovic, Jun 22 2004

Status

approved