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

Author

Alford Arnold, Jun 18 2004

Keywords

• nonn

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

Examples

			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
		

Links

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

Crossrefs

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

Programs

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 *)

PARI

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

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

Extensions

More terms from Vladeta Jovovic, Jun 22 2004