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A124233 Expansion of psi(q) * phi(-q^10) * chi(-q^5) / chi(-q^2) in powers of q where phi(), psi(), chi() are Ramanujan theta functions. 2
1, 1, 1, 2, 1, 1, 2, 2, 1, 3, 1, 0, 2, 0, 2, 2, 1, 0, 3, 0, 1, 4, 0, 2, 2, 1, 0, 4, 2, 2, 2, 0, 1, 0, 0, 2, 3, 0, 0, 0, 1, 2, 4, 2, 0, 3, 2, 2, 2, 3, 1, 0, 0, 0, 4, 0, 2, 0, 2, 0, 2, 2, 0, 6, 1, 0, 0, 2, 0, 4, 2, 0, 3, 0, 0, 2, 0, 0, 0, 0, 1, 5, 2, 2, 4, 0, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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

Number 38 of the 74 eta-quotients listed in Table I of Martin (1996).

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..10000

Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of eta(q^2) * eta(q^4) * eta(q^5) * eta(q^10) / (eta(q) * eta(q^20)) in powers of q.

Euler transform of period 20 sequence [ 1, 0, 1, -1, 0, 0, 1, -1, 1, -2, 1, -1, 1, 0, 0, -1, 1, 0, 1, -2, ...].

Moebius transform is period 20 sequence [ 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, ...].

a(n) is multiplicative with a(2^e) = a(5^e) = 1, a(p^e) = e+1 if p == 1, 3, 7, 9 (mod 20), a(p^e) = (1 + (-1)^e) / 2 if p == 11, 13, 17, 19 (mod 20).

G.f.: 1 + Sum_{k>0} x^k * (1 + x^(2*k)) * (1 + x^(6*k)) / (1 + x^(10*k)).

a(2*n) = a(5*n) = a(n), a(20*n + 11) = a(20*n + 13) = a(20*n + 17) = a(20*n + 19) = 0.

a(n) = A035170(n) unless n=0. a(2*n + 1) = A129390(n). a(4*n + 3) = 2 * A033764(n).

EXAMPLE

G.f. = 1 + q + q^2 + 2*q^3 + q^4 + q^5 + 2*q^6 + 2*q^7 + q^8 + 3*q^9 + q^10 + ...

MATHEMATICA

a[ n_] := If[ n < 1, Boole[n == 0], DivisorSum[ n, KroneckerSymbol[ -20, #] &]]; (* Michael Somos, Jul 09 2015 *)

a[ n_] := SeriesCoefficient[ QPochhammer[ q^2] QPochhammer[ q^4] QPochhammer[ q^5] QPochhammer[ q^10] / (QPochhammer[ q] QPochhammer[ q^20]), {q, 0, n}]; (* Michael Somos, Jul 09 2015 *)

PROG

(PARI) {a(n) = if( n<1, n==0, sumdiv( n, d, kronecker( -20, d)))};

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A) * eta(x^5 + A) * eta(x^10 + A) / eta(x + A) / eta(x^20 + A), n))};

CROSSREFS

Cf. A033764, A035170, A129390.

Sequence in context: A066888 A029313 A144001 * A035170 A111949 A143323

Adjacent sequences:  A124230 A124231 A124232 * A124234 A124235 A124236

KEYWORD

nonn,mult

AUTHOR

Michael Somos, Oct 21 2006

STATUS

approved

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Last modified June 18 00:21 EDT 2021. Contains 345088 sequences. (Running on oeis4.)