A308399 - OEIS

%I #22 Aug 05 2021 12:47:37

%S 1,0,0,1,0,1,1,0,2,1,1,3,1,3,3,2,6,3,4,8,4,9,9,6,15,10,12,20,12,22,23,

%T 18,35,26,30,46,32,51,54,45,76,62,71,99,76,111,117,104,160,136,154,

%U 205,167,230,244,223,319,286,319,406,349,456,484,458,619,570,632,779,695

%N Expansion of 1 / Sum_{k=-oo..oo} (-x)^(k*(4*k + 1)).

%C Number of partitions of n into parts congruent to {0, 3, 5} mod 8.

%C Convolution inverse of A244465.

%H Ludovic Schwob, <a href="/A308399/b308399.txt">Table of n, a(n) for n = 0..10000</a>

%F G.f.: 1 / Sum_{k>=0} (-x)^A074378(k).

%F G.f.: Product_{k>=1} 1 / ((1 - x^(8*k - 5)) * (1 - x^(8*k - 3)) * (1 - x^(8*k))).

%F G.f.: ( Sum_{k>=0} A000041(k)*(-x)^k ) / ( Sum_{k>=0} A000009(2*k)*(-x)^k ).

%F a(n) ~ sqrt(sqrt(2) - 1) * exp(sqrt(n)*Pi/2) / (2^(9/4)*n). - _Vaclav Kotesovec_, May 25 2019

%F a(n) = a(n-3) + a(n-5) - a(n-14) - a(n-18) + + - - (with a(0)=1 and a(n) = 0 for negative n), where 3, 5, 14, 18, ... is the sequence A074378. - _Ludovic Schwob_, Aug 04 2021

%e For n=23 the a(23)=6 solutions are 3+3+3+3+3+3+5, 3+3+3+3+3+8, 3+3+3+3+11, 3+5+5+5+5, 5+5+5+8, and 5+5+13.

%t nmax = 68; CoefficientList[Series[1/Sum[(-x)^(k (4 k + 1)), {k, -nmax, nmax}], {x, 0, nmax}], x]

%t nmax = 68; CoefficientList[Series[Product[1/((1 - x^(8 k - 5)) (1 - x^(8 k - 3)) (1 - x^(8 k))), {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 68; CoefficientList[Series[Sum[PartitionsP[k] (-x)^k, {k, 0, nmax}]/Sum[PartitionsQ[2 k] (-x)^k, {k, 0, nmax}], {x, 0, nmax}], x]

%Y Cf. A000009, A000041, A006950, A007742, A047622, A074378, A195850, A244465, A308400.

%K nonn

%O 0,9

%A _Ilya Gutkovskiy_, May 24 2019