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A261866 Expansion of (G(-x) / chi(-x))^2 in powers of x where chi() is a Ramanujan theta function and G() is a Rogers-Ramanujan function. 2
1, 0, 2, 2, 5, 6, 13, 16, 28, 38, 60, 80, 122, 162, 234, 312, 436, 576, 789, 1032, 1386, 1802, 2381, 3070, 4008, 5128, 6618, 8414, 10752, 13576, 17210, 21592, 27162, 33890, 42340, 52538, 65244, 80544, 99458, 122208, 150126, 183634, 224527, 273480, 332898 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Rogers-Ramanujan functions: G(q) (see A003114), H(q) (A003106).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Euler transform of period 20 sequence [ 0, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 0, 0, ...].
Expansion of (f(x^4, x^6) / f(-x^2, -x^3))^2 in powers of x where f(, ) is Ramanujan's general theta function.
Expansion of (f(-x, -x^9) * f(-x^8, -x^12) / (f(-x) * f(-x^20)))^2 in powers of x where f(, ) is Ramanujan's general theta function.
Expansion of (f(-x^2, x^3) / phi(-x^2))^2 in powers of x where phi() is a Ramanujan theta function.
G.f.: (Sum_{k>=0} x^(k^2 + k) / ((1 - x) * (1 - x^2) * ... * (1 - x^(2*k))))^2.
a(n) = A147699(5*n).
Convolution square of A122134.
EXAMPLE
G.f. = 1 + 2*x^2 + 2*x^3 + 5*x^4 + 6*x^5 + 13*x^6 + 16*x^7 + 28*x^8 + ...
G.f. = q + 2*q^41 + 2*q^61 + 5*q^81 + 6*q^101 + 13*q^121 + 16*q^141 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^4, -x^5] QPochhammer[ -x, -x^5] QPochhammer[ x, x^2])^-2, {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^-{0, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 0, 0}[[Mod[k, 20, 1]]], {k, n}], {x, 0, n}];
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^-[ 0, 0, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 0][k%20 + 1]), n))};
(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint(4*n + 1) - 1)\2, x^(k^2 + k) / prod(i=1, 2*k, 1 - x^i, 1 + x * O(x^(n - k^2-k))))^2, n))};
CROSSREFS
Sequence in context: A355021 A098600 A181716 * A147766 A034420 A304867
KEYWORD
nonn
AUTHOR
Michael Somos, Sep 03 2015
STATUS
approved

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Last modified July 18 10:00 EDT 2024. Contains 374378 sequences. (Running on oeis4.)