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

0,4

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).

Links

G. C. Greubel, Table of n, a(n) for n = 0..445

Eric Weisstein's MathWorld, Pochhammer Symbol.

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

Example

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 + ...

Mathematica

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

Prog

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

Crossrefs

Cf. A263687.

Sequence in context: A370569 A020072 A020027 * A197593 A084832 A135177

Adjacent sequences: A263685 A263686 A263687 * A263689 A263690 A263691

Keyword

nonn

Author

Vladimir Reshetnikov, Oct 23 2015

Status

approved