A263688 c(n) in (sqrt(2))_n = b(n) + c(n)*sqrt(2), where (x)_n is the Pochhammer symbol, b(n) and c(n) are integers.

0, 1, 1, 4, 18, 98, 630, 4676, 39368, 370748, 3861900, 44087008, 547360968, 7342948312, 105848450344, 1631635791184, 26782838577600, 466413214471568, 8588795078851344, 166747235206457024, 3404055687248777120, 72895914363584236064, 1633918325381940384864

Offset: 0

Author

Vladimir Reshetnikov, Oct 23 2015

Keywords

• nonn

Comments

The Pochhammer symbol (sqrt(2))_n = Gamma(n + sqrt(2))/Gamma(sqrt(2)) = sqrt(2)*(1 + sqrt(2))*(2 + sqrt(2))*...*(n - 1 + sqrt(2)).

(sqrt(2))_n = A263687(n) + a(n)*sqrt(2).

Examples

			For n = 4, (sqrt(2))_4 = sqrt(2)*(1 + sqrt(2))*(2 + sqrt(2))*(3 + sqrt(2)) = 26 + 18*sqrt(2), so a(4) = 18.
G.f. = x + x^2 + 4*x^3 + 18*x^4 + 98*x^5 + 630*x^6 + 4676*x^7 + 39368*x^8 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..445
• Eric Weisstein's MathWorld, Pochhammer Symbol.

Crossrefs

Cf. A263687.

Programs

Mathematica

Expand@Table[(Pochhammer[Sqrt[2], n] - Pochhammer[-Sqrt[2], n])/(2 Sqrt[2]), {n, 0, 22}]

PARI

{a(n) = if( n<0, 0, imag( prod(k=0, n-1, quadgen(8) + k)))}; /* Michael Somos, Oct 23 2015 */

Formula

a(n) = ((sqrt(2))_n - (-sqrt(2))_n)/(2*sqrt(2)).

E.g.f.: (1/(1-x)^sqrt(2)-(1-x)^sqrt(2))/(2*sqrt(2)) = -sinh(sqrt(2)*log(1-x))/sqrt(2).

D-finite with recurrence: a(0) = 0, a(1) = 1, a(n+2) = (2*n+1)*a(n+1) + (2-n^2)*a(n).

a(n) ~ exp(-n)*n^(n+sqrt(2)-1/2)*sqrt(Pi)/(2*Gamma(sqrt(2))).

0 = a(n)*(+7*a(n+1) - a(n+2) - 6*a(n+3) + a(n+4)) + a(n+1)*(+7*a(n+1) + 6*a(n+2) - 4*a(n+3)) + a(n+2)*(+3*a(n+2)) for all n>=0. - Michael Somos, Oct 23 2015