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A277431
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Expansion of e.g.f.: cosh(sqrt(2)*x)/(1-x).
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2
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1, 1, 4, 12, 52, 260, 1568, 10976, 87824, 790416, 7904192, 86946112, 1043353408, 13563594304, 189890320384, 2848354805760, 45573676892416, 774752507171072, 13945545129079808, 264965357452516352, 5299307149050328064, 111285450130056889344, 2448279902861251567616
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = (Gamma(n+1, sqrt(2))*exp(sqrt(2)) + Gamma(n+1, -sqrt(2))/exp(sqrt(2)))/2.
a(n) ~ sqrt(2*Pi)*cosh(sqrt(2))*n^(n+1/2)*exp(-n).
D-finite with recurrence: a(n) = n*a(n-1) + 2*a(n-2) - 2*(n-2)*a(n-3).
Gamma(n+1, sqrt(2))*exp(sqrt(2)) = a(n) + sqrt(2)*A277432(n).
Gamma(n+1, -sqrt(2))/exp(sqrt(2)) = a(n) - sqrt(2)*A277432(n).
a(2*n+1) = (2*n+1)*a(2*n).
0 = a(n)*(+4*a(n+1) -4*a(n+2) -6*a(n+3) +2*a(n+4)) +a(n+1)*(+4*a(n+1) +2*a(n+2) -4*a(n+3)) +a(n+2)*(+2*a(n+2) +a(n+3) -a(n+4)) + a(n+3)*(+a(n+3)) for all n>-3. - Michael Somos, Oct 01 2018
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EXAMPLE
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G.f. = 1 + x + 4*x^2 + 12*x^3 + 52*x^4 + 260*x^5 + 1568*x^6 + ... - Michael Somos, Oct 01 2018
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MATHEMATICA
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Round@Table[(Gamma[n + 1, Sqrt[2]] Exp[Sqrt[2]] + Gamma[n + 1, -Sqrt[2]]/Exp[Sqrt[2]])/2, {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster *)
Table[SeriesCoefficient[Cosh[Sqrt[2] x]/(1 - x), {x, 0, n}] n!, {n, 0, 20}]
a[ n_] := If[ n < 0, 0, n! Sum[ 2^k / (2 k)!, 0] {k, 0, n/2}]]; (* Michael Somos, Oct 01 2018 *)
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PROG
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(PARI) x='x+O('x^30); Vec(serlaplace(cosh(sqrt(2)*x)/(1-x))) \\ G. C. Greubel, Sep 30 2018
(PARI) {a(n) = if( n<0, 0, n! * sum(k=0, n\2, 2^k / (2*k)!))}; /* Michael Somos, Oct 01 2018 */
(Magma) I:=[1, 4, 12]; [1] cat [n le 3 select I[n] else n*Self(n-1) + 2*Self(n-2) - 2*(n-2)*Self(n-3): n in [1..30]]; // G. C. Greubel, Sep 30 2018
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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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