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A350896 Number of partitions of n such that 4*(smallest part) = (number of parts).

%I A350896 #25 Oct 14 2024 12:48:02
%S A350896 0,0,0,1,1,2,3,4,5,7,8,10,12,14,16,20,22,26,30,35,40,48,55,65,76,90,
%T A350896 105,126,147,175,206,244,286,339,396,467,545,638,741,865,1000,1160,
%U A350896 1337,1543,1770,2035,2325,2660,3029,3451,3916,4447,5029,5691,6419,7242,8146,9167,10286,11546,12930,14481,16185
%N A350896 Number of partitions of n such that 4*(smallest part) = (number of parts).
%H A350896 Vaclav Kotesovec, <a href="/A350896/b350896.txt">Table of n, a(n) for n = 1..10000</a>
%F A350896 G.f.: Sum_{k>=1} x^(4*k^2)/Product_{j=1..4*k-1} (1-x^j).
%F A350896 a(n) ~ c * exp(Pi*sqrt(2*n/5)) / n^(3/4), where c = (3 - sqrt(5))^(1/4) / (8*sqrt(5)) = 0.05226232058... - _Vaclav Kotesovec_, Jan 25 2022, updated Oct 13 2024
%e A350896 For n=7 there are a(7)=3 such partitions: [1,2,2,2], [1,1,2,3] and [1,1,1,4]. - _R. J. Mathar_, Jun 20 2022
%t A350896 CoefficientList[Series[Sum[x^(4k^2)/Product[1-x^j,{j,4k-1}],{k,63}],{x,0,63}],x] (* _Stefano Spezia_, Jan 22 2022 *)
%o A350896 (PARI) my(N=66, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, sqrtint(N\4), x^(4*k^2)/prod(j=1, 4*k-1, 1-x^j))))
%Y A350896 Column 4 of A350889.
%Y A350896 Cf. A168657.
%K A350896 nonn
%O A350896 1,6
%A A350896 _Seiichi Manyama_, Jan 21 2022