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A128617 Expansion of q^2 * psi(q) * psi(q^15) in powers of q where psi() is a Ramanujan theta function. 7
0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 1, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 2, 1, 0, 1, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,17

COMMENTS

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Also the number of positive odd solutions to equation x^2 + 15y^2 = 8n. - Seiichi Manyama, May 21 2017

REFERENCES

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 377, Entry 9(i).

LINKS

Seiichi Manyama, Table of n, a(n) for n = 1..10000

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of (eta(q^2) * eta(q^30))^2 / (eta(q) * eta(q^15)) in powers of q.

Euler transform of period 30 sequence [ 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 2, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -2, ...].

For n>0, n in A028955 equivalent to a(n) nonzero. If a(n) nonzero, a(n) = A082451(n) and a(n) = -A121362(n).

a(n)= (A082451(n) - A121362(n) )/2.

G.f.: x^2 * Product_{k>0} (1 - x^k) * (1 - x^(15*k)) * (1 + x^(2*k))^2 * (1 + x^(30*k))^2.

EXAMPLE

G.f. = x^2 + x^3 + x^5 + x^8 + x^12 + 2*x^17 + x^18 + x^20 + 2*x^23 + x^27 + x^30 + ...

MATHEMATICA

a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ -60, #] - KroneckerSymbol[ 20, #] KroneckerSymbol[ -3, n/#] &] / 2]; (* Michael Somos, Nov 12 2015 *)

a[ n_] := SeriesCoefficient[ q^2 (QPochhammer[ q^2] QPochhammer[ q^30])^2 / (QPochhammer[ q] QPochhammer[ q^15]), {q, 0, n}]; (* Michael Somos, Nov 12 2015 *)

PROG

(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, kronecker(-60, d) - kronecker(20, d) * kronecker(-3, n/d) )/2)};

(PARI) {a(n) = my(A); if( n<2, 0, n-=2; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^30 + A))^2 / (eta(x + A) * eta(x^15 + A)), n))};

CROSSREFS

Cf. A035162.

Sequence in context: A015738 A308103 A115604 * A116488 A216601 A283000

Adjacent sequences:  A128614 A128615 A128616 * A128618 A128619 A128620

KEYWORD

nonn

AUTHOR

Michael Somos, Mar 13 2007

STATUS

approved

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Last modified August 18 20:02 EDT 2019. Contains 326109 sequences. (Running on oeis4.)