A161090 Number of partitions of n into squares where every part appears at least 2 times.

0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4, 3, 5, 4, 6, 5, 6, 6, 7, 6, 8, 8, 9, 9, 11, 10, 13, 11, 14, 14, 16, 15, 18, 18, 20, 19, 22, 22, 25, 24, 27, 28, 32, 29, 36, 34, 39, 38, 42, 42, 47, 45, 51, 51, 56, 55, 62, 61, 68, 66, 75, 73, 82, 79, 88, 88, 96, 93, 104, 105, 112, 113, 122, 123

Offset: 1

Author

R. H. Hardin, Jun 02 2009

Keywords

• nonn

Examples

			a(12)=3 because we have 444, 441111, and 1^(12). - _Emeric Deutsch_, Jun 21 2009
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from R. H. Hardin)

Crossrefs

Cf. A001156, A103265.

Programs

Maple

g := -1+product(1+x^(2*j^2)/(1-x^(j^2)), j = 1 .. 10): gser := series(g, x = 0, 90): seq(coeff(gser, x, n), n = 1 .. 79); # Emeric Deutsch, Jun 21 2009

Mathematica

nmax = 100; Rest[CoefficientList[Series[-1 + Product[(1 + x^(2*k^2)/(1-x^(k^2))), {k, 1, Sqrt[nmax] + 1}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 15 2025 *)
nmax = 100; Rest[CoefficientList[Series[-1 + Product[(1 + x^(3*k^2))/(1 - x^(2*k^2)), {k, 1, Sqrt[nmax/2] + 1}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 15 2025 *)

Formula

G.f.: -1 + Product_{j>=1} (1 + x^(2*j^2)/(1-x^(j^2))). - Emeric Deutsch, Jun 21 2009

From Vaclav Kotesovec, Jun 15 2025: (Start)

G.f.: -1 + Product_{k>=1} (1 + x^(3*k^2)) / (1 - x^(2*k^2)).

a(n) ~ ((2 - sqrt(2) + sqrt(6))*zeta(3/2))^(2/3) * exp(Pi^(1/3)*(3*(2 - sqrt(2) + sqrt(6))*zeta(3/2))^(2/3)*n^(1/3)/4) / (8 * 3^(5/6) * Pi^(7/6) * n^(7/6)). (End)