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A129390 Expansion of phi(x) * phi(-x^5) / (chi(-x^2) * chi(-x^10)) in powers of x where phi(), chi() are Ramanujan theta functions. 4
1, 2, 1, 2, 3, 0, 0, 2, 0, 0, 4, 2, 1, 4, 2, 0, 0, 2, 0, 0, 2, 2, 3, 2, 3, 0, 0, 0, 0, 0, 2, 6, 0, 2, 4, 0, 0, 2, 0, 0, 5, 2, 0, 4, 2, 0, 0, 0, 0, 0, 2, 2, 4, 2, 2, 0, 0, 2, 0, 0, 1, 4, 1, 2, 4, 0, 0, 4, 0, 0, 4, 0, 2, 6, 2, 0, 0, 0, 0, 0, 4, 2, 0, 2, 1, 0, 0 (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
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
FORMULA
Expansion of q^(-1/2) * eta(q^2)^4 * eta(q^5)^2 * eta(q^20) / (eta(q)^2 * eta(q^4) * eta(q^10)^2) in powers of q.
Euler transform of period 20 sequence [ 2, -2, 2, -1, 0, -2, 2, -1, 2, -2, 2, -1, 2, -2, 0, -1, 2, -2, 2, -2, ...].
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0, b(5^e) = 1, b(p^e) = e+1 if p == 1, 3, 7, 9 (mod 20), b(p^e) = (1 + (-1)^e)/2 if p == 11, 13, 17, 19 (mod 20).
G.f.: Sum_{k>0} a(k) * x^(2*k - 1) = Sum_{k>0} f(x^(2*k - 1)) where f(x) := x * (1 + x^2) * (1 + x^6) / (1 + x^10).
a(n) = (-1)^n * A129391(n). a(n) = A035710(2*n + 1).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(5) = 1.404962... . - Amiram Eldar, Dec 28 2023
EXAMPLE
G.f. = 1 + 2*x + x^2 + 2*x^3 + 3*x^4 + 2*x^7 + 4*x^10 + 2*x^11 + x^12 + 4*x^13 + ...
G.f. = q + 2*q^3 + q^5 + 2*q^7 + 3*q^9 + 2*q^15 + 4*q^21 + 2*q^23 + q^25 + ...
MATHEMATICA
a[ n_] := If[ n < 0, 0, DivisorSum[ 2 n + 1, KroneckerSymbol[ -20, #]&]]; (* Michael Somos, Nov 12 2015 *)
PROG
(PARI) {a(n) = if( n<0, 0, n = 2*n + 1; sumdiv(n, d, kronecker( -20, d)))};
(PARI) {a(n) = my(A, p, e); if(n<0, 0, n = 2*n + 1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==5, 1, p%20 <10, e+1, 1-e%2) ))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^5 + A)^2 * eta(x^20 + A) / (eta(x + A)^2 * eta(x^4 + A) * eta(x^10 + A)^2), n))};
CROSSREFS
Sequence in context: A328362 A158810 A129391 * A345255 A348910 A123590
KEYWORD
nonn,easy
AUTHOR
Michael Somos, Apr 13 2007
STATUS
approved

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Last modified June 19 16:49 EDT 2024. Contains 373503 sequences. (Running on oeis4.)