A027412 a(n) = 2*a(n-1) + (n-2)*a(n-2) with a(0) = 1, a(1) = 2.

1, 2, 4, 10, 28, 86, 284, 998, 3700, 14386, 58372, 246218, 1076156, 4860710, 22635292, 108459814, 533813716, 2694524642, 13930068740, 73667056394, 398075350108, 2195824771702, 12353156545564, 70818633296870

Offset: 0

Author

Olivier Gérard

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• R. Donaghey, Binomial self-inverse sequences and tangent coefficients, J. Combin. Theory, Series A, 21 (1976), 155-163.

Crossrefs

Cf. A005425.

Programs

Magma

[n le 2 select n else 2*Self(n-1) + (n-3)*Self(n-2): n in [1..40]]; // G. C. Greubel, Apr 01 2023

Mathematica

RecurrenceTable[{a[0]==1, a[1]==2, a[n]==2 a[n-1] + (n-2) a[n-2]}, a,{ n, 40}] (* Vincenzo Librandi, Dec 29 2012 *)
Table[If[n==0, 1, 2*(-I/Sqrt[2])^(n-1)*HermiteH[n-1, I*Sqrt[2]]], {n, 0, 40}] (* G. C. Greubel, Apr 01 2023 *)

SageMath

[1]+[2*(-i/sqrt(2))^(n-1)*hermite(n-1, i*sqrt(2)) for n in range(1,41)] # G. C. Greubel, Apr 01 2023

Formula

a(n) = 2 * A005425(n-1), n > 0.

a(n) ~ sqrt(2)*exp(2*sqrt(n)-n/2-1)*n^(n/2-1/2). - Vaclav Kotesovec, Oct 08 2012