A277431 Expansion of e.g.f.: cosh(sqrt(2)*x)/(1-x).

1, 1, 4, 12, 52, 260, 1568, 10976, 87824, 790416, 7904192, 86946112, 1043353408, 13563594304, 189890320384, 2848354805760, 45573676892416, 774752507171072, 13945545129079808, 264965357452516352, 5299307149050328064, 111285450130056889344, 2448279902861251567616

Offset: 0

Author

Vladimir Reshetnikov, Oct 14 2016

Keywords

• nonn

Examples

			G.f. = 1 + x + 4*x^2 + 12*x^3 + 52*x^4 + 260*x^5 + 1568*x^6 + ... - _Michael Somos_, Oct 01 2018
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..448
• Eric Weisstein's MathWorld, Incomplete Gamma Function

Crossrefs

Cf. A000023, A263823, A277345, A277432.

Programs

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

Mathematica

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)!, {k, 0, n/2}]]; (* Michael Somos, Oct 01 2018 *)
With[{nn=30},CoefficientList[Series[Cosh[x Sqrt[2]]/(1-x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 13 2024 *)

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 */

Formula

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