A000807 Quadratic invariants.

1, 2, 14, 182, 3614, 99302, 3554894, 159175382, 8654995454, 558786468422, 42086200603694, 3645412584724022, 358877175474325214, 39758874175808713382, 4915216680878167372814, 673139563824188490513302, 101475126400695241802946494, 16744618803625299734467026182

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 0..100
• J. Touchard, Nombres exponentiels et nombres de Bernoulli, Canad. J. Math., 8 (1956), 305-320.

Crossrefs

Cf. A000110.

Programs

Maple

Bell := combinat:-bell:
A000807 := n -> add(binomial(2*n, k)*(-1)^k*Bell(k)*Bell(2*n-k), k = 0..2*n):
seq(A000807(n), n=0..17); # Peter Luschny, Sep 10 2017

Mathematica

nn = 40; t = Range[0, nn]! CoefficientList[Series[Exp[Exp[x] + Exp[-x] - 2], {x, 0, nn}], x]; Take[t, {1, nn, 2}] (* T. D. Noe, Jun 20 2012 *)

Python

from sympy import binomial, bell
def a(n): return sum(binomial(2*n, k)*(-1)**k*bell(k)*bell(2*n - k) for k in range(2*n  + 1))
print([a(n) for n in range(21)]) # Indranil Ghosh, Sep 11 2017

Formula

From Vladeta Jovovic, Sep 08 2002: (Start)

E.g.f.: exp(exp(x)+exp(-x)-2).

a(n) = Sum_{k=0..2*n} (-1)^k*binomial(2*n, k)*A000110(k)*A000110(2*n - k). (End)

a(0) = 1; a(n) = 2 * Sum_{k=1..n} binomial(2*n-1,2*k-1) * a(n-k). - Ilya Gutkovskiy, Jan 27 2020

Extensions

More terms from Vladeta Jovovic, Sep 08 2002