login
Quadratic invariants.
(Formerly M2071 N0819)
9

%I M2071 N0819 #34 May 02 2020 07:18:44

%S 1,2,14,182,3614,99302,3554894,159175382,8654995454,558786468422,

%T 42086200603694,3645412584724022,358877175474325214,

%U 39758874175808713382,4915216680878167372814,673139563824188490513302,101475126400695241802946494,16744618803625299734467026182

%N Quadratic invariants.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

%H T. D. Noe, <a href="/A000807/b000807.txt">Table of n, a(n) for n = 0..100</a>

%H J. Touchard, <a href="http://dx.doi.org/10.4153/CJM-1956-034-1">Nombres exponentiels et nombres de Bernoulli</a>, Canad. J. Math., 8 (1956), 305-320.

%F From _Vladeta Jovovic_, Sep 08 2002: (Start)

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

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

%F 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

%p Bell := combinat:-bell:

%p A000807 := n -> add(binomial(2*n, k)*(-1)^k*Bell(k)*Bell(2*n-k), k = 0..2*n):

%p seq(A000807(n), n=0..17); # _Peter Luschny_, Sep 10 2017

%t 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 *)

%o (Python)

%o from sympy import binomial, bell

%o def a(n): return sum(binomial(2*n, k)*(-1)**k*bell(k)*bell(2*n - k) for k in range(2*n + 1))

%o print([a(n) for n in range(21)]) # _Indranil Ghosh_, Sep 11 2017

%Y Cf. A000110.

%K nonn

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _Vladeta Jovovic_, Sep 08 2002