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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.

%I A263687 #25 Feb 16 2025 08:33:27
%S A263687 1,0,2,6,26,140,896,6636,55804,525168,5468008,62403880,774616696,
%T A263687 10390122288,149757486368,2308301709840,37887797229968,
%U A263687 659770432834688,12148923787132832,235858218326093664,4814800618608693664,103104123746671427520,2310978427407268450048
%N 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.
%C A263687 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 A263687 (sqrt(2))_n = a(n) + A263688(n)*sqrt(2).
%H A263687 G. C. Greubel, <a href="/A263687/b263687.txt">Table of n, a(n) for n = 0..445</a>
%H A263687 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PochhammerSymbol.html">Pochhammer Symbol</a>.
%F A263687 a(n) = ((sqrt(2))_n + (-sqrt(2))_n)/2.
%F A263687 E.g.f.: (1/(1-x)^sqrt(2)+(1-x)^sqrt(2))/2 = cosh(sqrt(2)*log(1-x)).
%F A263687 Recurrence: a(0) = 1, a(1) = 0, a(n+2) = (2*n+1)*a(n+1) + (2-n^2)*a(n).
%F A263687 a(n) ~ exp(-n)*n^(n+sqrt(2)-1/2)*sqrt(Pi/2)/Gamma(sqrt(2)).
%F A263687 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
%F A263687 From _Benedict W. J. Irwin_, Oct 14 2016: (Start)
%F A263687 a(n) = (-1)^n*(binomial(-sqrt(2), n) + binomial(sqrt(2), n))*n!/2.
%F A263687 Conjecture: a(n) = (-1)^n * Sum_{k=0..n/2} Stirling1(n,2*k)*2^k.
%F A263687 (End)
%e A263687 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.
%e A263687 G.f. = 1 + 2*x^2 + 6*x^3 + 26*x^4 + 140*x^5 + 896*x^6 + 6636*x^7 + 55804*x^8 + ...
%t A263687 Expand@Table[(Pochhammer[Sqrt[2], n] + Pochhammer[-Sqrt[2], n])/2, {n, 0, 22}]
%o A263687 (PARI) {a(n) = if( n<0, 0, real(prod(k=0, n-1, quadgen(8) + k)))}; /* _Michael Somos_, Oct 23 2015 */
%Y A263687 Cf. A263688.
%K A263687 nonn
%O A263687 0,3
%A A263687 _Vladimir Reshetnikov_, Oct 23 2015