A321234 Denominator of series expansion of the hypergeometric series 3F2([1/2, 1, 1], [3/2, 3/2], x).

1, 9, 75, 245, 2835, 7623, 39039, 96525, 1859715, 4387955, 20369349, 46646691, 422524375, 947754675, 4217257575, 9316746045, 327288272355, 714666904875, 3105965056425, 6720018279975, 57930003736605, 124404851229945, 532600050191625, 1136728029829275, 19356624110780775

Offset

0,2

Links

Table of n, a(n) for n=0..24.

Formula

a(n) = numerator(binomial(2*n, n)/4^n) * (2*n+1)^2. - G. C. Greubel, Dec 07 2018

From Peter Bala, Apr 27 2024: (Start)

[x^n] 3F2([1/2, 1, 1], [3/2, 3/2], x) = 1/(2*n + 1) * Integral_{x = 0..1} (1 - x^2)^n dx = 1/(2*n + 1) * Sum_{k = 0..n} (-1)^k * binomial(n, k)/(2*k + 1).

a(n) = denominator of 2^(2*n+1)/ ((n + 1)*(2*n + 1)*binomial(2*n+2, n+1)). (End)

Maple

a:=n->(2*n+1)^2*binomial(2*n, n)/4^n: seq(numer(a(n)), n=0..25); # Muniru A Asiru, Dec 08 2018

Mathematica

Denominator[CoefficientList[Series[HypergeometricPFQ[{1/2, 1, 1}, {3/2, 3/2}, c], {c, 0, 20}], c]]
Table[(2*n+1)^2*Numerator[Binomial[2*n, n]/4^n], {n, 0, 30}] (* G. C. Greubel, Dec 07 2018 *)

Prog

(PARI) vector(30, n, n--; numerator(binomial(2*n, n)/4^n)*(2*n+1)^2) \\ G. C. Greubel, Dec 07 2018
(Magma) [Numerator(Binomial(2*n, n)/4^n)*(2*n+1)^2: n in [0..30]]; // G. C. Greubel, Dec 07 2018
(Sage) [numerator(binomial(2*n, n)/4^n)*(2*n+1)^2 for n in range(30)] # G. C. Greubel, Dec 07 2018
(GAP) List([0..30], n -> NumeratorRat(Binomial(2*n, n)/4^n)*(2*n+1)^2); # G. C. Greubel, Dec 07 2018

Crossrefs

Numerators appear to be A046161.

Sequence in context: A028991 A249396 A102094 * A339483 A274311 A281804

Adjacent sequences: A321231 A321232 A321233 * A321235 A321236 A321237

Keyword

nonn,frac

Author

Eugene d'Eon, Nov 01 2018

Status

approved