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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.

%I A263688 #17 Feb 16 2025 08:33:27
%S A263688 0,1,1,4,18,98,630,4676,39368,370748,3861900,44087008,547360968,
%T A263688 7342948312,105848450344,1631635791184,26782838577600,466413214471568,
%U A263688 8588795078851344,166747235206457024,3404055687248777120,72895914363584236064,1633918325381940384864
%N 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.
%C A263688 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)).
%C A263688 (sqrt(2))_n = A263687(n) + a(n)*sqrt(2).
%H A263688 G. C. Greubel, <a href="/A263688/b263688.txt">Table of n, a(n) for n = 0..445</a>
%H A263688 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/PochhammerSymbol.html">Pochhammer Symbol</a>.
%F A263688 a(n) = ((sqrt(2))_n - (-sqrt(2))_n)/(2*sqrt(2)).
%F A263688 E.g.f.: (1/(1-x)^sqrt(2)-(1-x)^sqrt(2))/(2*sqrt(2)) = -sinh(sqrt(2)*log(1-x))/sqrt(2).
%F A263688 D-finite with recurrence: a(0) = 0, a(1) = 1, a(n+2) = (2*n+1)*a(n+1) + (2-n^2)*a(n).
%F A263688 a(n) ~ exp(-n)*n^(n+sqrt(2)-1/2)*sqrt(Pi)/(2*Gamma(sqrt(2))).
%F A263688 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
%e A263688 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.
%e A263688 G.f. = x + x^2 + 4*x^3 + 18*x^4 + 98*x^5 + 630*x^6 + 4676*x^7 + 39368*x^8 + ...
%t A263688 Expand@Table[(Pochhammer[Sqrt[2], n] - Pochhammer[-Sqrt[2], n])/(2 Sqrt[2]), {n, 0, 22}]
%o A263688 (PARI) {a(n) = if( n<0, 0, imag( prod(k=0, n-1, quadgen(8) + k)))}; /* _Michael Somos_, Oct 23 2015 */
%Y A263688 Cf. A263687.
%K A263688 nonn
%O A263688 0,4
%A A263688 _Vladimir Reshetnikov_, Oct 23 2015