cp's OEIS Frontend

A363601 Number of partitions of n where there are k^2 - 1 kinds of parts k.

%I A363601 #27 Jan 16 2025 11:29:56
%S A363601 1,0,3,8,21,48,126,288,693,1568,3570,7896,17417,37632,80823,171192,
%T A363601 359733,747936,1543192,3155760,6407037,12909024,25835649,51359136,
%U A363601 101470854,199264128,389096028,755591256,1459643343,2805471984,5366161740,10216161336,19362398580
%N A363601 Number of partitions of n where there are k^2 - 1 kinds of parts k.
%F A363601 G.f.: 1/Product_{k>=1} (1-x^k)^(k^2-1).
%F A363601 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} A092348(k) * a(n-k).
%F A363601 G.f.: exp(Sum_{k >= 1} (sigma_3(k) - sigma_1(k))*x^k/k) = 1 + 3*x^2 + 8*x^3 + 21*x^4 + 48*x^5 + .... - _Peter Bala_, Jan 16 2025
%p A363601 with(numtheory):
%p A363601 series(exp(add((sigma[3](k) - sigma[1](k))*x^k/k, k = 1..50)), x, 51):
%p A363601 seq(coeftayl(%, x = 0, n), n = 0..50); # _Peter Bala_, Jan 16 2025
%o A363601 (PARI) my(N=40, x='x+O('x^N)); Vec(1/prod(k=1, N, (1-x^k)^(k^2-1)))
%Y A363601 Cf. A005380, A023871, A052847, A092348, A120844, A217093, A253289, A255271, A255802, A255803, A255835, A255836, A363602.
%K A363601 nonn,easy
%O A363601 0,3
%A A363601 _Seiichi Manyama_, Jun 10 2023
%E A363601 Name suggested by _Joerg Arndt_, Jun 11 2023