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A129303 Expansion of eta(q^2)^3 * eta(q^5)^2 * eta(q^10) / eta(q)^2 in powers of q. 4
1, 2, 2, 4, 5, 4, 6, 8, 7, 10, 12, 8, 12, 12, 10, 16, 16, 14, 20, 20, 12, 24, 22, 16, 25, 24, 20, 24, 30, 20, 32, 32, 24, 32, 30, 28, 36, 40, 24, 40, 42, 24, 42, 48, 35, 44, 46, 32, 43, 50, 32, 48, 52, 40, 60, 48, 40, 60, 60, 40, 62, 64, 42, 64, 60, 48, 66, 64 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

REFERENCES

S. Cooper, On Ramanujan's function k(q)=r(q)r^2(q^2), Ramanujan J., 20 (2009), 311-328; see p. 318 Th. 4.1

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 1..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of q * psi(q)^3 * psi(q^5) - q^2 * psi(q) * psi(q^5)^3 in powers of q where psi() is a Ramanujan theta function. - Michael Somos, Jul 12 2012

Euler transform of period 10 sequence [ 2, -1, 2, -1, 0, -1, 2, -1, 2, -4, ...].

a(n) is multiplicative with a(p^e) = p^e if p = 2 or 5, a(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 9 (mod 10), a(p^e) = (p^(e+1) + (-1)^e) / (p + 1) if p == 3, 7 (mod 10).

G.f.: Sum_{k>0} Kronecker(20, k) * x^k / (1 - x^k)^2.

G.f.: x * Product_{k>0} (1 - x^k) * (1 + x^(5*k)) * (1 + x^k)^3 * (1 - x^(5*k))^3.

a(2*n) = a(n). a(2*n + 1) = A134080(n).

EXAMPLE

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

MATHEMATICA

a[ n_] := If[ n < 1, 0, DivisorSum[ n, n/# KroneckerSymbol[ 20, #] &]]; (* Michael Somos, Jul 12 2012 *)

a[ n_] := SeriesCoefficient[ (1/16) (EllipticTheta[ 2, 0, q]^3 EllipticTheta[ 2, 0, q^5] - EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^5]^3), {q, 0, 2 n}]; (* Michael Somos, Jul 12 2012 *)

nmax = 100; Rest[CoefficientList[Series[x * Product[(1 - x^k) * (1 + x^(5*k)) * (1 + x^k)^3 * (1 - x^(5*k))^3, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Sep 08 2015 *)

PROG

(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, n/d * kronecker( 20, 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( 20, 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)^3 * eta(x^5 + A)^2 * eta(x^10 + A) / eta(x + A)^2, n))};

CROSSREFS

Cf. A134080.

Sequence in context: A293974 A346036 A138557 * A255368 A186101 A284722

Adjacent sequences: A129300 A129301 A129302 * A129304 A129305 A129306

KEYWORD

nonn,mult

AUTHOR

Michael Somos, Apr 08 2007

STATUS

approved

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Last modified November 29 18:13 EST 2022. Contains 358431 sequences. (Running on oeis4.)