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A259771
Expansion of x * psi(x^5) * f(-x^10) / f(-x^2,-x^8) in powers of x where psi(), f() are Ramanujan theta functions.
1
1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 3, 3, 4, 4, 5, 4, 6, 5, 7, 7, 9, 8, 10, 10, 12, 12, 15, 14, 18, 17, 20, 20, 24, 24, 28, 28, 33, 33, 38, 38, 44, 45, 50, 52, 59, 60, 68, 69, 78, 80, 89, 92, 102, 105, 116, 120, 133, 137, 151, 156, 171, 178, 194, 201
OFFSET
1,9
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
The g.f. for this sequence is the last term of the 14th equation on page 20 of Ramanujan 1988.
REFERENCES
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 20
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Euler transform of period 10 sequence [ 0, 1, 0, 0, 1, 0, 0, 1, 0, -1, ...].
a(n) = A053265(n-1) - A053267(n).
EXAMPLE
G.f. = x + x^3 + x^5 + x^6 + x^7 + x^8 + 2*x^9 + x^10 + 2*x^11 + x^12 + ...
G.f. = q^49 + q^289 + q^529 + q^649 + q^769 + q^889 + 2*q^1009 + q^1129 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ x Product[ (1 - x^k)^{ 0, -1, 0, 0, -1, 0, 0, -1, 0, 1}[[Mod[k, 10, 1]]], {k, n}], {x, 0, n}];
QP:= QPochhammer; a[n_]:= SeriesCoefficient[ x*QP[x^10]/(QP[x^5, x^10]* QP[x^2, x^10]*QP[x^8, x^10]), {x, 0, n}]; Table[a[n], {n, 1, 100}] (* G. C. Greubel, Mar 16 2018 *)
PROG
(PARI) {a(n) = if( n<1, 0, n--; polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 1, 0, -1, 0, 0, -1, 0, 0, -1, 0][k%10 + 1]), n))};
CROSSREFS
Sequence in context: A212813 A112219 A035458 * A357457 A194902 A194874
KEYWORD
nonn
AUTHOR
Michael Somos, Jul 04 2015
STATUS
approved