cp's OEIS Frontend

A145467 Convolution square of A003114.

%I A145467 #16 Aug 19 2020 06:16:27
%S A145467 1,2,3,4,7,10,15,20,28,38,52,68,91,118,153,196,252,318,403,504,632,
%T A145467 784,973,1196,1473,1800,2198,2668,3238,3908,4714,5660,6789,8112,9683,
%U A145467 11516,13685,16210,19178,22628,26671,31354,36821,43140,50489,58968,68796
%N A145467 Convolution square of A003114.
%H A145467 Georg Fischer, <a href="/A145467/b145467.txt">Table of n, a(n) for n = 0..1000</a>
%F A145467 a(n) = A145466(5*n).
%F A145467 Expansion of G(x)^2 in powers of x where G() is a Rogers-Ramanujan function.
%F A145467 a(n) ~ exp(2*Pi*sqrt(2*n/15)) * phi / (3^(1/4) * 10^(3/4) * n^(3/4)) * (1 - (3*sqrt(15/2)/(16*Pi) + Pi/(15*sqrt(30)))/sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Aug 14 2018
%F A145467 Euler transform of the period 5 sequence [2, 0, 0, 2, 0, ...]. - _Georg Fischer_, Aug 19 2020
%e A145467 1/q + 2*q^29 + 3*q^59 + 4*q^89 + 7*q^119 + 10*q^149 + 15*q^179 + ...
%p A145467 # Using the function EULER from Transforms (see link at the bottom of the page).
%p A145467 [1,op(EULER([seq(op([2,0,0,2,0]),n=1..9)]))]; # _Peter Luschny_, Aug 19 2020
%p A145467 # second Maple program:
%p A145467 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A145467       a(n-j)*add(`if`(irem(d, 5) in {1, 4}, 2*d, 0),
%p A145467        d=numtheory[divisors](j)), j=1..n)/n)
%p A145467     end:
%p A145467 seq(a(n), n=0..50);  # _Alois P. Heinz_, Aug 19 2020
%t A145467 nmax = 50; CoefficientList[Series[Product[1/((1 - x^(5*k - 1))*(1 - x^(5*k - 4)))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 14 2018 *)
%o A145467 (PARI) {a(n) = local(t); if( n<0, 0, t = 1 + x * O(x^n); polcoeff( sum(k=1, sqrtint(n), t *= x^(2*k - 1) / (1 - x^k) * (1 + x * O(x^(n - k^2))), 1)^2, n))}
%Y A145467 Cf. A003114, A145466, A001622.
%K A145467 nonn
%O A145467 0,2
%A A145467 _Michael Somos_, Oct 11 2008