A376813 G.f.: Sum_{k>=0} x^(k*(k+1)) * Product_{j=1..k} (1 + x^j)^2.

1, 0, 1, 2, 1, 0, 1, 2, 3, 4, 3, 2, 2, 2, 3, 6, 7, 8, 10, 8, 8, 8, 6, 8, 10, 12, 16, 20, 22, 24, 27, 26, 25, 26, 25, 26, 29, 32, 37, 44, 52, 58, 66, 72, 76, 82, 82, 84, 87, 88, 91, 96, 103, 112, 126, 138, 154, 174, 190, 208, 225, 238, 253, 268, 275, 284, 296, 304

Offset

0,4

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Formula

G.f.: Sum_{k>=0} Product_{j=1..k} (x^j + x^(2*j))^2.

a(n) ~ phi^(1/2) * exp(Pi*sqrt(2*n/15)) / (4 * 5^(1/4) * sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

Mathematica

nmax = 100; CoefficientList[Series[Sum[x^(n*(n+1))*Product[1+x^k, {k, 1, n}]^2, {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^k)*(1 + x^k)*x^(2*k)]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p; , {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]

Crossrefs

Cf. A333179, A376623, A376812, A216222.

Sequence in context: A245615 A053646 A080776 * A360659 A297158 A065358

Adjacent sequences: A376810 A376811 A376812 * A376814 A376815 A376816

Keyword

nonn

Author

Vaclav Kotesovec, Oct 05 2024

Status

approved