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A263687
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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.
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7
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1, 0, 2, 6, 26, 140, 896, 6636, 55804, 525168, 5468008, 62403880, 774616696, 10390122288, 149757486368, 2308301709840, 37887797229968, 659770432834688, 12148923787132832, 235858218326093664, 4814800618608693664, 103104123746671427520, 2310978427407268450048
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OFFSET
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0,3
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COMMENTS
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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).
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LINKS
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FORMULA
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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
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)
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EXAMPLE
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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 + ...
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MATHEMATICA
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Expand@Table[(Pochhammer[Sqrt[2], n] + Pochhammer[-Sqrt[2], n])/2, {n, 0, 22}]
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PROG
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(PARI) {a(n) = if( n<0, 0, real(prod(k=0, n-1, quadgen(8) + k)))}; /* Michael Somos, Oct 23 2015 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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