A263687 b(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.

1, 0, 2, 6, 26, 140, 896, 6636, 55804, 525168, 5468008, 62403880, 774616696, 10390122288, 149757486368, 2308301709840, 37887797229968, 659770432834688, 12148923787132832, 235858218326093664, 4814800618608693664, 103104123746671427520, 2310978427407268450048

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 = a(n) + A263688(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) = 26.
G.f. = 1 + 2*x^2 + 6*x^3 + 26*x^4 + 140*x^5 + 896*x^6 + 6636*x^7 + 55804*x^8 + ...
		

Links

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

Crossrefs

Cf. A263688.

Programs

Mathematica

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

PARI

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

Formula

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

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

Recurrence: a(0) = 1, a(1) = 0, 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

From Benedict W. J. Irwin, Oct 14 2016: (Start)

a(n) = (-1)^n*(binomial(-sqrt(2), n) + binomial(sqrt(2), n))*n!/2.

Conjecture: a(n) = (-1)^n * Sum_{k=0..n/2} Stirling1(n,2*k)*2^k.

(End)