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A128616 Expansion of q * psi(q^3) * psi(q^5) in powers of q where psi() is a Ramanujan theta function. 2
1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 1, 0, 0, 0, 2, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,19

COMMENTS

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

REFERENCES

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

LINKS

G. C. Greubel, 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^6) * eta(q^10))^2 / (eta(q^3) * eta(q^5)) in powers of q.

Euler transform of period 30 sequence [ 0, 0, 1, 0, 1, -1, 0, 0, 1, -1, 0, -1, 0, 0, 2, 0, 0, -1, 0, -1, 1, 0, 0, -1, 1, 0, 1, 0, 0, -2, ...].

For n>0, n in A028957 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 * Product_{k>0} (1 - x^(3*k)) * (1 - x^(5*k)) * (1 + x^(6*k))^2 * (1 + x^(10*k))^2.

EXAMPLE

G.f. = x + x^4 + x^6 + x^9 + x^10 + x^15 + x^16 + 2*x^19 + x^24 + x^25 + 2*x^31 + ...

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(QPochhammer[ q^6] QPochhammer[ q^10])^2 / (QPochhammer[ q^3] QPochhammer[ q^5]), {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<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^6 + A) * eta(x^10 + A))^2 / (eta(x^3 + A) * eta(x^5 + A)), n))};

CROSSREFS

Cf. A082451, A121362.

Sequence in context: A245536 A291203 A256852 * A270417 A307666 A319995

Adjacent sequences:  A128613 A128614 A128615 * A128617 A128618 A128619

KEYWORD

nonn

AUTHOR

Michael Somos, Mar 13 2007

STATUS

approved

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Last modified July 20 11:28 EDT 2019. Contains 325180 sequences. (Running on oeis4.)