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A124815 Expansion of q * psi(q)^2 * psi(-q^3)^2 * phi(-q^6) / phi(-q^2) in powers of q where phi(), psi() are Ramanujan theta functions. 6
1, 2, 3, 4, 4, 6, 6, 8, 9, 8, 12, 12, 14, 12, 12, 16, 16, 18, 18, 16, 18, 24, 24, 24, 21, 28, 27, 24, 28, 24, 30, 32, 36, 32, 24, 36, 38, 36, 42, 32, 40, 36, 42, 48, 36, 48, 48, 48, 43, 42, 48, 56, 52, 54, 48, 48, 54, 56, 60, 48, 62, 60, 54, 64, 56, 72, 66, 64, 72, 48, 72, 72 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Number 24 of the 74 eta-quotients listed in Table I of Martin (1996).

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

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..5000

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^3) / eta(q))^2 * eta(q^4) * eta(q^12) in powers of q.

Euler transform of period 12 sequence [ 2, 0, 0, -1, 2, -2, 2, -1, 0, 0, 2, -4, ...].

a(n) is multiplicative with a(p^e) = p^e if p<5, a(p^e) = (p^(e+1) - 1) / (p-1) if p == 1, 11 (mod 12), a(p^e) = (p^(e+1) + (-1)^e) / (p+1) if p == 5, 7 (mod 12).

G.f.: Sum_{k>0} k * x^k * (1 - x^(2*k)) / (1 - x^(2*k) + x^(4*k)).

G.f.: x * Product_{k>0} (1 + x^k)^2 * (1 - x^(3*k))^2 * (1 - x^(4*k)) * (1 - x^(12*k)).

a(2*n) = 2 * a(n).

EXAMPLE

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

MATHEMATICA

a[ n_] := If[ n < 1, 0, Sum[ n/d KroneckerSymbol[ 12, d], { d, Divisors[ n]}]]; (* Michael Somos, Jul 09 2015 *)

a[ n_] := SeriesCoefficient[ q QPochhammer[ q^2]^2 QPochhammer[ q^3]^2 QPochhammer[ q^4] QPochhammer[ q^12]/QPochhammer[ q]^2, {q, 0, n}]; (* Michael Somos, Jul 09 2015 *)

PROG

(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, n/d * kronecker( 12, d)))};

(PARI) {a(n) = my(A, p, e, f); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; f = kronecker( 12, p); (p^(e+1) - f^(e+1)) / (p - f)))};

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^4 + A) * eta(x^12 + A) / eta(x + A)^2, n))};

CROSSREFS

Sequence in context: A102443 A102441 A102440 * A081328 A179276 A213635

Adjacent sequences:  A124812 A124813 A124814 * A124816 A124817 A124818

KEYWORD

nonn,mult

AUTHOR

Michael Somos, Nov 08 2006

STATUS

approved

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Last modified October 28 17:59 EDT 2021. Contains 348329 sequences. (Running on oeis4.)