A376228 a(n) = (6*n+1) * (2*n)!^3 / n!^6.

1, 56, 2808, 152000, 8575000, 496093248, 29188893888, 1738242215424, 104455598247000, 6321316756040000, 384702925005146176, 23520160000755565056, 1443504313932496274368, 88879637239345064000000, 5487711609457595160000000, 339644002672064899081728000, 21065385579274083203741943000

Offset: 0

Author

Paul D. Hanna, Oct 17 2024

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 56*x + 2808*x^2 + 152000*x^3 + 8575000*x^4 + 496093248*x^5 + 29188893888*x^6 + 1738242215424*x^7 + ...
where
A(x) = 1 + 7*(1/2)^3*64*x + 13*((1*3)/(2*4))^3*64^2*x^2 + 19*((1*3*5)/(2*4*6))^3*64^3*x^3 + 25*((1*3*5*7)/(2*4*6*8))^3*64^4*x^4 + ... + (6*n+1)*(2*n)!^3/n!^6*x^n + ...
SPECIFIC VALUES.
At x = 1/256 we have the series
4/Pi = 1 + 7*(1/2)^3/4 + 13*((1*3)/(2*4))^3/4^2 + 19*((1*3*5)/(2*4*6))^3/4^3 + 25*((1*3*5*7)/(2*4*6*8))^3/4^4 + ... = 1.273239544735162686...
see formula 28 in the Ramanujan link for details.
		

Links

• S. Ramanujan, Modular equations and approximations to Pi, Quarterly Journal of Mathematics, XLV, 1914, p. 45.

Crossrefs

Cf. A002897.

Programs

Mathematica

a[n_]:=(6*n+1) * (2*n)!^3 / n!^6; Array[a,17,0] (* Stefano Spezia, Oct 17 2024 *)

Formula

a(n) = (6*n+1) * A002897(n).

a(n) ~ 3*2^(6*n+1)/sqrt(n*Pi^3). - Stefano Spezia, Oct 17 2024

D-finite with recurrence n^3*a(n) +8*(56*n^3-252*n^2+330*n-141)*a(n-1) -4096*(2*n-3)^3*a(n-2)=0. - R. J. Mathar, Oct 24 2024