A263002 Expansion of (f(-x^5) / f(-x))^2 in powers of x where f() is a Ramanujan theta function.

1, 2, 5, 10, 20, 34, 61, 100, 165, 260, 408, 620, 940, 1390, 2045, 2960, 4257, 6040, 8525, 11900, 16522, 22738, 31130, 42300, 57210, 76872, 102834, 136800, 181230, 238900, 313725, 410160, 534330, 693330, 896655, 1155420, 1484274, 1900420, 2426215, 3088100

Offset

0,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Number of 5-regular bipartitions of n. - N. J. A. Sloane, Oct 20 2019

References

Kathiravan, T., and S. N. Fathima. "On L-regular bipartitions modulo L." The Ramanujan Journal 44.3 (2017): 549-558.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1/3) * (eta(q^5) / eta(q))^2 in powers of q.

Euler transform of period 5 sequence [ 2, 2, 2, 2, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (45 t)) = (1/5) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A058511.

Given g.f. A(x), then B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u - v^2) * (v - u^2) - 4*u^2*v^2.

Convolution inverse is A058511.

a(n) ~ exp(4*Pi*sqrt(n/15)) / (sqrt(2) * 3^(1/4) * 5^(5/4) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015

See Maple code for a simple g.f. - N. J. A. Sloane, Oct 20 2019

Example

G.f. = 1 + 2*x + 5*x^2 + 10*x^3 + 20*x^4 + 34*x^5 + 61*x^6 + 100*x^7 + ...
G.f. = q + 2*q^4 + 5*q^7 + 10*q^10 + 20*q^13 + 34*q^16 + 61*q^19 + 100*q^22 + ...

Maple

f:=(k, M) -> mul(1-q^(k*j), j=1..M);
LRBP := (L, M) -> (f(L, M)/f(1, M))^2;
S := L -> seriestolist(series(LRBP(L, 80), q, 60));
S(5); # N. J. A. Sloane, Oct 20 2019

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ x^5] / QPochhammer[ x])^2, {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^5 + A) / eta(x + A))^2, n))};

Crossrefs

Cf. A058511.

Number of r-regular bipartitions of n for r = 2,3,4,5,6: A022567, A328547, A001936, A263002, A328548.

Sequence in context: A365630 A126105 A117486 * A325649 A325719 A000710

Adjacent sequences: A262999 A263000 A263001 * A263003 A263004 A263005

Keyword

nonn

Author

Michael Somos, Oct 07 2015

Status

approved