A360487 - OEIS

%I #7 Feb 09 2023 10:45:03

%S 0,1,5,14,31,60,106,176,279,426,631,912,1291,1795,2457,3317,4424,5837,

%T 7626,9875,12684,16171,20476,25764,32228,40094,49626,61131,74966,

%U 91545,111346,134921,162906,196031,235134,281175,335251,398615,472695,559115,659721,776608

%N Convolution of A000009 and A000290.

%F a(n) = Sum_{k=0..n} A000009(k) * (n-k)^2.

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

%F a(n) ~ 4 * 3^(5/4) * n^(3/4) * exp(sqrt(n/3)*Pi) / Pi^3.

%p b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(

%p      `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)

%p     end:

%p a:= n-> add(b(n-j)*j^2, j=0..n):

%p seq(a(n), n=0..42);  # _Alois P. Heinz_, Feb 09 2023

%t Table[Sum[PartitionsQ[k]*(n-k)^2, {k, 0, n}], {n, 0, 60}]

%t CoefficientList[Series[x*(1+x)*QPochhammer[-1, x] / (2*(1-x)^3), {x, 0, 60}], x]

%Y Cf. A000009, A000290, A095944, A360486.

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Feb 09 2023