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A093831
Expansion of q * (chi(-q) * chi(-q^5))^-4 in powers of q where chi() is a Ramanujan theta function.
3
1, 4, 10, 24, 51, 104, 206, 384, 697, 1228, 2112, 3568, 5898, 9592, 15358, 24256, 37850, 58340, 88980, 134344, 200972, 298112, 438538, 640256, 928041, 1336104, 1911436, 2717776, 3842110, 5401784, 7555012, 10514176, 14562432, 20077672
OFFSET
1,2
COMMENTS
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
Expansion of (eta(q^2) * eta(q^10) / (eta(q) * eta(q^5)))^4 in powers of q.
Euler transform of period 10 sequence [ 4, 0, 4, 0, 8, 0, 4, 0, 4, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v*(1 + 8*u + 16*u*v).
G.f.: x * (Product_{k>0} (1 - x^(10*k - 5)) * (1 - x^(2*k - 1)))^-4.
Convolution inverse of A132040. - Michael Somos, Apr 26 2015
a(n) ~ exp(2*Pi*sqrt(2*n/5)) / (16 * 2^(3/4) * 5^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015
EXAMPLE
G.f. = q + 4*q^2 + 10*q^3 + 24*q^4 + 51*q^5 + 104*q^6 + 206*q^7 + 384*q^8 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ q (QPochhammer[ -q, q] QPochhammer[ -q^5, q^5] )^4, {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)
nmax = 40; Rest[CoefficientList[Series[x * Product[1/((1 - x^(10*k - 5)) * (1 - x^(2*k - 1)))^4, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Sep 07 2015 *)
PROG
(PARI) {a(n) = if( n<1, 0, n--; polcoeff( (1 / prod(k=1, (n+5)\10, 1 - x^(10*k - 5), 1 + x * O(x^n)) / prod(k=1, (n+1)\2, 1 - x^(2*k - 1), 1 + x * O(x^n)))^4, n))};
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^10 + A) / (eta(x + A) * eta(x^5 + A)))^4, n))};
CROSSREFS
Cf. A132040.
Sequence in context: A309777 A128516 A022569 * A274582 A052365 A107659
KEYWORD
nonn
AUTHOR
Michael Somos, Apr 17 2004, Oct 04 2004
EXTENSIONS
Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 05 2007
STATUS
approved