|
|
A145468
|
|
Convolution square of A003106.
|
|
1
|
|
|
1, 0, 2, 2, 3, 4, 7, 8, 13, 16, 23, 28, 40, 48, 66, 82, 107, 132, 171, 208, 266, 324, 406, 494, 614, 740, 912, 1098, 1338, 1604, 1945, 2318, 2793, 3320, 3972, 4706, 5605, 6612, 7840, 9222, 10882, 12760, 15004, 17534, 20542, 23944, 27949, 32490, 37813, 43832
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
LINKS
|
|
|
FORMULA
|
Expansion of H(x)^2 in powers of x where H() is a Rogers-Ramanujan function.
Euler transform of period 5 sequence [ 0, 2, 2, 0, 0, ...].
G.f.: (1 + Sum_{k>0} x^(k^2 - k) / ((1 - t) * (1 - t^2) * ... * (1 - t^k)))^2 = (Product_{k>0} (1 - x^(5*k - 2)) * (1 - x^(5*k -3)))^-2.
a(n) ~ (sqrt(5)-1) * exp(2*Pi*sqrt(2*n/15)) / (2^(7/4) * 3^(1/4) * 5^(3/4) * n^(3/4)) * (1 + (11*Pi/(15*sqrt(30)) - 3*sqrt(15/2)/(16*Pi)) / sqrt(n)). - Vaclav Kotesovec, Aug 14 2018
|
|
EXAMPLE
|
q^11 + 2*q^71 + 2*q^101 + 3*q^131 + 4*q^161 + 7*q^191 + 8*q^221 + ...
|
|
MATHEMATICA
|
nmax = 50; CoefficientList[Series[Product[1/((1 - x^(5*k - 2))*(1 - x^(5*k - 3)))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 14 2018 *)
|
|
PROG
|
(PARI) {a(n) = local(t); if( n<0, 0, t = 1 + x * O(x^n); polcoeff( sum(k= 1, (sqrt(4*n + 1) - 1) / 2, t *= x^(2*k) / (1 - x^k) * (1 + x * O(x^(n - k^2 - k))), 1)^2, n))}
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|