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A134080 Expansion of (f(-q^5)^5 / f(-q) + f(q^5)^5 / f(q)) / 2 in powers of q^2 where f() is a Ramanujan theta function. 3
1, 2, 5, 6, 7, 12, 12, 10, 16, 20, 12, 22, 25, 20, 30, 32, 24, 30, 36, 24, 42, 42, 35, 46, 43, 32, 52, 60, 40, 60, 62, 42, 60, 66, 44, 72, 72, 50, 72, 80, 61, 82, 80, 60, 90, 72, 64, 100, 96, 84, 102, 102, 60, 106, 110, 72, 112, 110, 84, 96, 133, 84, 125, 126 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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 = 0..10000

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of ( phi(x^5) * psi(x^2) + x * phi(x) * psi(x^10) ) * f(-x^5) * phi(-x^5) / chi(-x) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.

a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(5^e) = 5^e, b(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 9 (mod 10), b(p^e) = (p^(e+1)  + (-1)^e) / (p + 1) if p == 3, 7 (mod 10).

a(n) = A053723(2*n) = A110712(2*n + 1) = A129303(2*n + 1) = A138483(2*n + 1) = A138512(2*n + 1) = A138557(2*n + 1).

EXAMPLE

G.f. = 1 + 2*x + 5*x^2 + 6*x^3 + 7*x^4 + 12*x^5 + 12*x^6 + 10*x^7 + 16*x^8 + ...

G.f. = q + 2*q^3 + 5*q^5 + 6*q^7 + 7*q^9 + 12*q^11 + 12*q^13 + 10*q^15 + 16*q^17 + ...

MATHEMATICA

a[ n_] := With[ {m = 2 n + 1}, If[ m < 1, 0, Sum[ m/d KroneckerSymbol[ 5, d], {d, Divisors @ m}]]]; (* Michael Somos, Jun 14 2014 *)

PROG

(PARI) {a(n) = if( n<0, 0, n = 2*n + 1 ; sumdiv(n, d, kronecker( 5, d) * n / d)) };

CROSSREFS

Cf. A053723, A110712, A129303, A138483, A138512, A138557.

Sequence in context: A058605 A282183 A103025 * A111300 A215761 A117548

Adjacent sequences:  A134077 A134078 A134079 * A134081 A134082 A134083

KEYWORD

nonn

AUTHOR

Michael Somos, Oct 07 2007

STATUS

approved

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Last modified September 21 22:17 EDT 2019. Contains 327284 sequences. (Running on oeis4.)