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Counting a set of restricted partitions.
6

%I #23 Oct 09 2024 21:33:00

%S 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,

%T 29,30,32,35,40,45,53,60,67,73,79,85,87,92,95,98,105,111,120,132,145,

%U 160,178,196,212,231,247,263,280,291,305,319,334,352,371,393

%N Counting a set of restricted partitions.

%H Vaclav Kotesovec, <a href="/A216222/b216222.txt">Table of n, a(n) for n = 0..10000</a>

%F 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+...

%F 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

%t Take[CoefficientList[Sum[x^(k^2)*Product[1 + x^i, {i, k}]^2, {k, 0, 7}], x], 63] (* _Giovanni Resta_, Mar 13 2013 *)

%t 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]

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

%K nonn

%O 0,3

%A _David S. Newman_, Mar 13 2013

%E a(14)-a(62) from _Giovanni Resta_, Mar 13 2013