A350894 - OEIS

%I #19 Jan 24 2022 09:13:45

%S 0,0,1,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,5,5,

%T 6,6,7,7,8,8,9,9,10,10,11,11,12,13,14,15,17,18,20,22,24,26,29,31,34,

%U 37,40,43,47,50,54,58,62,66,71,75,80,85,90,95,102,107,114,121,129,136,146,154,165,175,187,198,213

%N Number of partitions of n such that (smallest part) = 3*(number of parts).

%H Vaclav Kotesovec, <a href="/A350894/b350894.txt">Table of n, a(n) for n = 1..10000</a>

%F G.f.: Sum_{k>=1} x^(3*k^2)/Product_{j=1..k-1} (1-x^j).

%F a(n) ~ (1 - alfa) * exp(2*sqrt(n*(3*log(alfa)^2 + polylog(2, 1 - alfa)))) * (3*log(alfa)^2 + polylog(2, 1 - alfa))^(1/4) / (2*sqrt(Pi) * sqrt(6 - 5*alfa) * n^(3/4)), where alfa = 0.7780895986786010978806823096592944458720784440255... is positive real root of the equation alfa^6 + alfa - 1 = 0. - _Vaclav Kotesovec_, Jan 22 2022

%o (PARI) my(N=99, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, sqrtint(N\3), x^(3*k^2)/prod(j=1, k-1, 1-x^j))))

%Y Column 3 of A350890.

%Y Cf. A168656, A237756.

%K nonn

%O 1,27

%A _Seiichi Manyama_, Jan 21 2022