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A255065
Expansion of x * psi(x^5) * f(-x^10) / f(-x^4, -x^6) in powers of x where psi(), f() are Ramanujan theta functions.
1
1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4, 4, 4, 4, 6, 5, 6, 6, 8, 8, 10, 9, 11, 11, 13, 13, 16, 15, 17, 18, 21, 21, 24, 24, 28, 29, 32, 33, 38, 38, 43, 44, 49, 51, 57, 58, 65, 67, 73, 76, 85, 87, 95, 99, 109, 113, 123, 127, 139, 145, 157
OFFSET
1,13
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Rogers-Ramanujan functions: G(q) (see A003114), H(q) (A003106).
REFERENCES
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 23, 7th equation.
LINKS
FORMULA
Expansion of x * f(-x, -x^9) * f(-x^10) / f(-x, -x^4) in powers of x where f(,) is the Ramanujan general theta function.
Expansion of x * psi(x^5) * H(x^2) in powers of x where f(,) is the Ramanujan general theta function and H() is a Rogers-Ramanujan function. - Michael Somos, Jul 09 2015
Euler transform of period 10 sequence [ 0, 0, 0, 1, 1, 1, 0, 0, 0, -1, ...].
G.f.: x * (Sum_{k>0} x^(5*k*(k-1)/2)) / (Product_{k in Z} 1 - x^abs(10*k + 4)).
A053266(n) = A053264(n) + a(n) unless n=0.
EXAMPLE
G.f. = x + x^5 + x^6 + x^7 + x^9 + x^10 + x^11 + x^12 + 2*x^13 + x^14 + ...
G.f. = q^119 + q^599 + q^719 + q^839 + q^1079 + q^1199 + q^1319 + q^1439 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ Product[ x * (1 - x^k)^{ 0, 0, 0, -1, -1, -1, 0, 0, 0, 1} [[Mod[k, 10, 1]]], {k, n}], {x, 0, n}];
PROG
(PARI) {a(n) = if( n<1, 0, n--; polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 1, 0, 0, 0, -1, -1, -1, 0, 0, 0][k%10+1]), n))};
CROSSREFS
Sequence in context: A161090 A349219 A178697 * A027349 A053277 A078661
KEYWORD
nonn
AUTHOR
Michael Somos, Jul 08 2015
STATUS
approved