A274655 Numerators of coefficients of z^n for the expansion of Fricke's hypergeometric function F_1(1/2,1/2;z).

0, 1, 21, 185, 18655, 102501, 1394239, 33944053, 3074289075, 99205524275, 7190934788323, 4590859955591, 2435122285235549, 23468182735812325, 38870446014205425, 145536272272236993, 280137373064011153371, 1633533514217325226737, 74200692627870055029475

Offset: 0

Author

Wolfdieter Lang, Jul 07 2016

Keywords

• nonn
• easy
• frac

Comments

For the denominators see A274656.

The main entry is A274653/A274654. In A274653 Fricke's hypergeometric function F_1(a,b;z) is defined by the recurrence. More details and the Fricke references are also found there.

Examples

			The rationals R(n) begin:
0, 1/2, 21/64, 185/768, 18655/98304, 102501/655360, 1394239/10485760, 33944053/293601280, ...
		

Crossrefs

Cf. A274656, A274653/A274654.

Formula

a(n) = numerator(R(n)), where the rationals (in lowest terms) are R(n) = [z^n]F_1(1/2,1/2;z), and the recurrence for R(n) = r(n)/n! is obtained from the one given for r(n) in A274653.

R(n) = ((2*n-1)/(2*n))^2*R(n-1) + 2*C(n)/(n*(2*n-1)), n >= 1, R(0) = 0, with C(n) = ((2*n)!)^2 / (n!^4*2^(4*n)).