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
Keywords
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Programs
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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