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

1, 1, 1, 2, 3, 3, 5, 6, 8, 12, 12, 17, 23, 27, 32, 41, 52, 61, 77, 91, 110, 134, 159, 188, 228, 271, 314, 380, 444, 518, 612, 713, 832, 976, 1128, 1308, 1529, 1756, 2023, 2343, 2698, 3091, 3555, 4072, 4657, 5343, 6074, 6922, 7912, 8986, 10194, 11590, 13135, 14855

Offset

0,4

Comments

Also the number of partitions of n in which each part occurs a square number (>=0) of times.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Seiichi Manyama)

Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018, p. 30.

Formula

G.f.: Product_{k>=1} (theta_3(x^k) + 1)/2, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Oct 25 2018

Example

n | Partitions of n in which each part occurs a square number (>=0) of times
--+-------------------------------------------------------------------------
1 | 1;
2 | 2;
3 | 3 = 2+1;
4 | 4 = 3+1 = 1+1+1+1;
5 | 5 = 4+1 = 3+2;
6 | 6 = 5+1 = 4+2 = 3+2+1 = 2+1+1+1+1;
7 | 7 = 6+1 = 5+2 = 4+3 = 4+2+1 = 3+1+1+1+1;
8 | 8 = 7+1 = 6+2 = 5+3 = 5+2+1 = 4+3+1 = 4+1+1+1+1 = 2+2+2+2;

Maple

b:= proc(n, i) option remember; local j; if n=0 then 1
      elif i<1 then 0 else b(n, i-1); for j while
        i*j^2<=n do %+b(n-i*j^2, i-1) od; % fi
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, May 11 2018

Mathematica

nmax = 60; CoefficientList[Series[Product[(EllipticTheta[3, 0, x^k] + 1)/2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 25 2018 *)

Prog

(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
(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

Crossrefs

Cf. A000041, A000122, A010052, A158441, A232173, A298329, A304329, A320932, A321139.

Sequence in context: A339405 A039860 A084338 * A039876 A239312 A317167

Adjacent sequences: A300443 A300444 A300445 * A300447 A300448 A300449

Keyword

nonn

Author

Seiichi Manyama, May 11 2018

Status

approved