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%I #35 Dec 07 2020 11:21:31
%S 1,3,8,30,133,768,5221,41302,369170,3677058,40338310,483134179,
%T 6271796072,87709287104,1314511438945,21017751750506,357102350816602,
%U 6424883282375340,122025874117476166,2439726373093186274
%N Row sums for triangle A096162.
%C 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
%H Seiichi Manyama, <a href="/A096161/b096161.txt">Table of n, a(n) for n = 1..449</a>
%F G.f.: B(x)*B(x^2)*B(x^3)*... where B(x) is g.f. of A000142. - _Christian G. Bower_, Jan 17 2006
%F G.f.: Product_{k>0} Sum_{r>=0} x^(r*k)*r!. - _Andrew Howroyd_, Dec 22 2017
%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_, Aug 10 2019
%e 1 1 2 1 3 6 1 4 6 12 24 ... A036038
%e 1 1 1 1 3 1 1 4 3 6 1 ... A036040
%e 1 1 2 1 1 6 1 1 2 2 24 ... A096162
%e so a(n) begins 1 3 8 30 ... A096161
%t 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 *)
%t 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 *)
%o (PARI) { my(n=25); Vec(prod(k=1, n, O(x*x^n) + sum(r=0, n\k, x^(r*k)*r!))) }
%Y Cf. A005651, A000110, A036038, A036040, A096162, A110143, A287899.
%K nonn
%O 1,2
%A _Alford Arnold_, Jun 18 2004
%E More terms from _Vladeta Jovovic_, Jun 22 2004
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