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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. 5
0, 1, 1, 4, 18, 98, 630, 4676, 39368, 370748, 3861900, 44087008, 547360968, 7342948312, 105848450344, 1631635791184, 26782838577600, 466413214471568, 8588795078851344, 166747235206457024, 3404055687248777120, 72895914363584236064, 1633918325381940384864 (list; graph; refs; listen; history; text; internal format)
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: A054139 A020072 A020027 * A197593 A084832 A135177

Adjacent sequences:  A263685 A263686 A263687 * A263689 A263690 A263691

KEYWORD

nonn

AUTHOR

Vladimir Reshetnikov, Oct 23 2015

STATUS

approved

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Last modified August 12 09:24 EDT 2020. Contains 336438 sequences. (Running on oeis4.)