A000807 Quadratic invariants. (Formerly M2071 N0819)

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

Offset

0,2

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.

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

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 *)

Prog

(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

Crossrefs

Cf. A000110.

Sequence in context: A230991 A258872 A372246 * A191565 A191236 A217905

Adjacent sequences: A000804 A000805 A000806 * A000808 A000809 A000810

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Sep 08 2002

Status

approved