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A263002
Expansion of (f(-x^5) / f(-x))^2 in powers of x where f() is a Ramanujan theta function.
6
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
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
KEYWORD
nonn
AUTHOR
Michael Somos, Oct 07 2015
STATUS
approved