OFFSET
0,2
LINKS
Christian Krattenthaler and Thomas W. Müller, The congruence properties of Romik's sequence of Taylor coefficients of Jacobi's theta function theta_3, arXiv:2304.11471 [math.NT], 2023. See p. 5.
FORMULA
a(n) = Product_{j=1..n} (4*j - 1)^2 - Sum_{m=0..n-1} binomial(2*n+1, 2*m+1)*Product_{j=1..n-m} (4*j - 3)^2*a(m) for n > 0.
MATHEMATICA
Table[(2n+1)!SeriesCoefficient[x*Hypergeometric2F1[3/4, 3/4, 3/2, 4*x^2]/Hypergeometric2F1[1/4, 1/4, 1/2, 4*x^2], {x, 0, 2n+1}], {n, 0, 14}]
(* or *)
a[0]=1; a[n_]:=Product[(4j-1)^2, {j, n}]-Sum[Binomial[2n+1, 2m+1]Product[(4j-3)^2, {j, n-m}]a[m], {m, 0, n-1}]; Array[a, 15, 0]
CROSSREFS
KEYWORD
nonn
AUTHOR
Stefano Spezia, Apr 30 2023
STATUS
approved