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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A274654 Denominators of coefficients of z^n/n! for the expansion of Fricke's hypergeometric function F_1(1/2,1/2;z).

Original entry on oeis.org

1, 2, 32, 128, 4096, 16384, 131072, 524288, 33554432, 134217728, 1073741824, 4294967296, 68719476736, 274877906944, 2199023255552, 8796093022208, 1125899906842624, 4503599627370496, 36028797018963968, 144115188075855872, 2305843009213693952
Offset: 0

Views

Author

Wolfdieter Lang, Jul 07 2016

Keywords

Comments

The numerators are given in A274653, where one finds the definition of Fricke's F1(a,b;z) by a recurrence and references.

Examples

			See A274653.

References

Crossrefs

Cf. A274653.

Formula

a(n) = denominator(r(n)), with the rationals (in lowest terms) defined by the recurrence
r(n) = ((2*n-1)^2/(4*n))*r(n-1) + 2*c(n)/(n*(2*n-1)), n >= 1, r(0) = 0, with c(n) = ((2*n)!)^2 / (n!^3*2^(4*n)).

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

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, Jul 07 2016

Keywords

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)).

A274656 Denominators of coefficients of z^n for the expansion of Fricke's hypergeometric function F_1(1/2,1/2;z).

Original entry on oeis.org

1, 2, 64, 768, 98304, 655360, 10485760, 293601280, 30064771072, 1082331758592, 86586540687360, 60473139527680, 34832528367943680, 362258295026614272, 644014746713980928, 2576058986855923712, 5275768805080931762176, 32613843522318487257088
Offset: 0

Views

Author

Wolfdieter Lang, Jul 07 2016

Keywords

Comments

For the numerators see A274655.
For the denominators of the coefficients of z^n/n! for the expansion of F_1(1/2,1/2;z) see A274654.
See the main entry A274653 (with A274654) for the definition of Fricke's hypergeometric function F_1(a,b;z) with the recurrence and details on F_1(1/2,1/2;z).

Examples

			See A274653, A274654, A274655.

References

Crossrefs

Cf. A274653.

Formula

a(n) = denominator(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)).
Showing 1-3 of 3 results.

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