A216222 Counting a set of restricted partitions.

1, 1, 2, 1, 1, 2, 3, 4, 3, 3, 3, 3, 6, 7, 8, 10, 9, 9, 9, 9, 11, 13, 16, 20, 22, 25, 28, 27, 28, 29, 30, 32, 35, 40, 45, 53, 60, 67, 73, 79, 85, 87, 92, 95, 98, 105, 111, 120, 132, 145, 160, 178, 196, 212, 231, 247, 263, 280, 291, 305, 319, 334, 352, 371, 393

Offset: 0

Author

David S. Newman, Mar 13 2013

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A001622, A306734, A376542, A376580, A376812, A376813.

Programs

Mathematica

Take[CoefficientList[Sum[x^(k^2)*Product[1 + x^i, {i, k}]^2, {k, 0, 7}], x], 63] (* Giovanni Resta, Mar 13 2013 *)
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^k)*(1 + x^k)*x^(2*k - 1)]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p; , {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1] (* Vaclav Kotesovec, Oct 09 2024 *)

Formula

G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^j)^2 = 1 +x^1*(1+x)^2 +x^4*(1+x)^2*(1+x^2)^2 +...+ x^k^2*(1+x)^2*(1+x^2)^2*(1+x^3)^2*...*(1+x^k)^2+...

a(n) ~ phi^(3/2) * exp(Pi*sqrt(2*n/15)) / (4*5^(1/4)*sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 29 2024

Extensions

a(14)-a(62) from Giovanni Resta, Mar 13 2013