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A281541 Expansion of Sum_{i>=1} x^(i^2)/(1 - x^(i^2)) / Product_{j>=1} (1 - x^(j^2)). 8

%I #13 Sep 15 2021 02:24:28

%S 1,2,3,5,7,9,11,15,19,23,27,34,41,47,53,64,75,86,96,113,129,145,159,

%T 182,206,229,252,284,318,349,380,423,468,513,555,616,676,736,791,869,

%U 949,1026,1103,1202,1310,1408,1506,1631,1766,1896,2020,2185,2354,2525,2680,2882,3094,3305,3506,3751,4023,4281,4537

%N Expansion of Sum_{i>=1} x^(i^2)/(1 - x^(i^2)) / Product_{j>=1} (1 - x^(j^2)).

%C Total number of parts in all partitions of n into squares.

%C Convolution of A001156 and A046951.

%H Alois P. Heinz, <a href="/A281541/b281541.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>

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

%F a(n) = Sum_{k=1..n} k * A243148(n,k). - _Alois P. Heinz_, Sep 19 2018

%F a(n) ~ exp(3 * 2^(-4/3) * zeta(3/2)^(2/3) * (Pi*n)^(1/3)) * sqrt(Pi/3) / (12*sqrt(n)). - _Vaclav Kotesovec_, Sep 15 2021

%e a(8) = 15 because we have [4, 4], [4, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1] and 2 + 5 + 8 = 15.

%p b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n], (s->

%p `if`(s>n, 0, (p->p+[0, p[1]])(b(n-s, i))))(i^2)+b(n, i-1))

%p end:

%p a:= n-> b(n, isqrt(n))[2]:

%p seq(a(n), n=1..70); # _Alois P. Heinz_, Sep 19 2018

%t nmax = 63; Rest[CoefficientList[Series[Sum[x^i^2/(1 - x^i^2), {i, 1, nmax}]/Product[1 - x^j^2, {j, 1, nmax}], {x, 0, nmax}], x]]

%Y Cf. A000290, A001156, A006128, A046951, A243148.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Jan 23 2017

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Last modified March 29 06:57 EDT 2024. Contains 371265 sequences. (Running on oeis4.)