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A209452 a(n) = Pell(n)*A122859(n) for n>=1, with a(0)=1, where A122859 lists the coefficients in phi(-q)^3/phi(-q^3) and phi() is a Ramanujan theta function. 4
1, -6, 24, -30, -72, 0, 840, -2028, 4896, -5910, 0, 0, -83160, -401532, 1938768, 0, -2824992, 0, 32930520, -79501308, 0, -463367580, 0, 0, 6520076640, -7870428726, 76003583088, -45872220270, -221490672624, 0, 0, -3116610274188, 7524162792576, 0, 0, 0, -127800022137480 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Compare the g.f. to the Lambert series of A122859: 1 - 6*Sum_{n>=1} Kronecker(n,3)*x^n/(1+x^n).

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

FORMULA

G.f.: 1 - 6*Sum_{n>=1} Pell(n)*Kronecker(n,3)*x^n/(1 + A002203(n)*x^n + (-1)^n*x^(2*n)), where A002203(n) = Pell(n-1) + Pell(n+1).

EXAMPLE

G.f.: A(x) = 1 - 6*x + 24*x^2 - 30*x^3 - 72*x^4 + 840*x^6 - 2028*x^7 + ...

where A(x) = 1 - 1*6*x + 2*12*x^2 - 5*6*x^3 - 12*6*x^4 + 70*12*x^6 - 169*12*x^7 + 408*12*x^8 - 985*6*x^9 + ... + Pell(n)*A122859(n)*x^n + ...

The g.f. is also given by the identity:

A(x) = 1 - 6*( 1*x/(1+2*x-x^2) - 2*x^2/(1+6*x^2+x^4) + 12*x^4/(1+34*x^4+x^8) - 29*x^5/(1+82*x^5-x^10) + 169*x^7/(1+478*x^7-x^14) - 408*x^8/(1+1154*x^8+x^16) + ...).

The values of the symbol Kronecker(n,3) repeat [1,-1,0, ...].

MATHEMATICA

A122859[n_]:= SeriesCoefficient[EllipticTheta[4, 0, q]^3/EllipticTheta[4, 0, q^3], {q, 0, n}]; Join[{1}, Table[Fibonacci[n, 2]*A122859[n], {n, 1, 50}]] (* G. C. Greubel, Jan 02 2017 *)

PROG

(PARI) {Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)), n)}

{A002203(n)=Pell(n-1)+Pell(n+1)}

{a(n)=polcoeff(1 - 6*sum(m=1, n, Pell(m)*kronecker(m, 3)*x^m/(1+A002203(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))), n)}

for(n=0, 40, print1(a(n), ", "))

CROSSREFS

Cf. A122859, A205972, A209451, A209453, A209446, A209449, A204270, A000129 (Pell), A002203.

Sequence in context: A132398 A073120 A147778 * A275302 A201596 A128459

Adjacent sequences:  A209449 A209450 A209451 * A209453 A209454 A209455

KEYWORD

sign

AUTHOR

Paul D. Hanna, Mar 10 2012

STATUS

approved

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Last modified February 24 01:16 EST 2020. Contains 332195 sequences. (Running on oeis4.)