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A300446 Expansion of Product_{k>0} (Sum_{m>=0} x^(k*m^2)).

%I A300446 #74 Sep 08 2022 08:46:20
%S A300446 1,1,1,2,3,3,5,6,8,12,12,17,23,27,32,41,52,61,77,91,110,134,159,188,
%T A300446 228,271,314,380,444,518,612,713,832,976,1128,1308,1529,1756,2023,
%U A300446 2343,2698,3091,3555,4072,4657,5343,6074,6922,7912,8986,10194,11590,13135,14855
%N A300446 Expansion of Product_{k>0} (Sum_{m>=0} x^(k*m^2)).
%C A300446 Also the number of partitions of n in which each part occurs a square number (>=0) of times.
%H A300446 Alois P. Heinz, <a href="/A300446/b300446.txt">Table of n, a(n) for n = 0..10000</a> (first 1001 terms from Seiichi Manyama)
%H A300446 Martin Klazar, <a href="http://arxiv.org/abs/1808.08449">What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I</a>, arXiv:1808.08449 [math.CO], 2018, p. 30.
%F A300446 G.f.: Product_{k>=1} (theta_3(x^k) + 1)/2, where theta_3() is the Jacobi theta function. - _Ilya Gutkovskiy_, Oct 25 2018
%e A300446 n | Partitions of n in which each part occurs a square number (>=0) of times
%e A300446 --+-------------------------------------------------------------------------
%e A300446 1 | 1;
%e A300446 2 | 2;
%e A300446 3 | 3 = 2+1;
%e A300446 4 | 4 = 3+1 = 1+1+1+1;
%e A300446 5 | 5 = 4+1 = 3+2;
%e A300446 6 | 6 = 5+1 = 4+2 = 3+2+1 = 2+1+1+1+1;
%e A300446 7 | 7 = 6+1 = 5+2 = 4+3 = 4+2+1 = 3+1+1+1+1;
%e A300446 8 | 8 = 7+1 = 6+2 = 5+3 = 5+2+1 = 4+3+1 = 4+1+1+1+1 = 2+2+2+2;
%p A300446 b:= proc(n, i) option remember; local j; if n=0 then 1
%p A300446       elif i<1 then 0 else b(n, i-1); for j while
%p A300446         i*j^2<=n do %+b(n-i*j^2, i-1) od; % fi
%p A300446     end:
%p A300446 a:= n-> b(n$2):
%p A300446 seq(a(n), n=0..60);  # _Alois P. Heinz_, May 11 2018
%t A300446 nmax = 60; CoefficientList[Series[Product[(EllipticTheta[3, 0, x^k] + 1)/2, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Oct 25 2018 *)
%o A300446 (PARI) N=99; x='x+O('x^N); Vec(prod(i=1, N, sum(j=0, sqrtint(N\i), x^(i*j^2)))) \\ _Seiichi Manyama_, Oct 28 2018
%o A300446 (Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(&+[x^(k*j^2):j in [0..2*m]]): k in [1..2*m]]) ));  // _G. C. Greubel_, Oct 29 2018
%Y A300446 Cf. A000041, A000122, A010052, A158441, A232173, A298329, A304329, A320932, A321139.
%K A300446 nonn
%O A300446 0,4
%A A300446 _Seiichi Manyama_, May 11 2018