A204342 a(n) = (-1)^n * Sum_{2*m + 1 | 2*n + 1} (-1)^m (2*m + 1)^4.
1, 80, 626, 2400, 6481, 14640, 28562, 50080, 83522, 130320, 192000, 279840, 391251, 524960, 707282, 923520, 1171200, 1502400, 1874162, 2284960, 2825762, 3418800, 4057106, 4879680, 5762401, 6681760, 7890482, 9164640, 10425600
Offset: 0
Examples
1 + 80*x + 626*x^2 + 2400*x^3 + 6481*x^4 + 14640*x^5 + 28562*x^6 + ... q + 80*q^3 + 626*q^5 + 2400*q^7 + 6481*q^9 + 14640*q^11 + 28562*q^13 + ... a(1) = 80 since (-1)^1 * ( (-1)^0 * 1^4 + (-1)^1 * 3^4) = 80 where 1 and 3 are the odd divisors of 3 = 2*1 + 1.
References
- L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 315.
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Michael Somos, Introduction to Ramanujan theta functions, 2019.
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
Programs
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Mathematica
QP:= QPochhammer[q]; a[n_]:= SeriesCoefficient[QP[q^2]^14* (QP[q]^8 + 80*q*QP[q^4]^8)/(QP[q]^8*QP[q^4]^4), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Apr 11 2018 *) -
PARI
{a(n) = if( n<0, 0, (-1)^n * sumdiv( 2*n + 1, d, (-1)^(d\2) * d^4))} -
PARI
{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^14 * (eta(x + A)^8 + 80 * x * eta(x^4 + A)^8) / (eta(x + A)^8 * eta(x^4 + A)^4), n))}
Formula
Expansion of phi(x)^4 * psi(x^2)^2 * (phi(x)^4 + 64 * x * psi(x^2)^4) in powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of q^(-1/2) * eta(q^2)^14 * (eta(q)^8 + 80 * q * eta(q^4)^8) / (eta(q)^8 * eta(q^4)^4) in powers of q.
a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = ((p^4)^(e+1) + 1) / (p^4 + 1) if p == 3 (mod 4), b(p^e) = ((p^4)^(e+1) - 1) / (p^4 - 1) if p == 1 (mod 4).
G.f.: Sum_{k > 0} (2*k - 1)^4 * x^(2*k - 1) / (1 + x^(4*k - 2)).
a(n) = A050468(2*n + 1).
Sum_{k=1..n} a(k) ~ c * n^5, where c = Pi^5/96 = 3.187705... . - Amiram Eldar, Dec 29 2023
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