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A000807 Quadratic invariants.

%I A000807 M2071 N0819 #34 May 02 2020 07:18:44
%S A000807 1,2,14,182,3614,99302,3554894,159175382,8654995454,558786468422,
%T A000807 42086200603694,3645412584724022,358877175474325214,
%U A000807 39758874175808713382,4915216680878167372814,673139563824188490513302,101475126400695241802946494,16744618803625299734467026182
%N A000807 Quadratic invariants.
%D A000807 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000807 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000807 T. D. Noe, <a href="/A000807/b000807.txt">Table of n, a(n) for n = 0..100</a>
%H A000807 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 A000807 From _Vladeta Jovovic_, Sep 08 2002: (Start)
%F A000807 E.g.f.: exp(exp(x)+exp(-x)-2).
%F A000807 a(n) = Sum_{k=0..2*n} (-1)^k*binomial(2*n, k)*A000110(k)*A000110(2*n - k). (End)
%F A000807 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 A000807 Bell := combinat:-bell:
%p A000807 A000807 := n -> add(binomial(2*n, k)*(-1)^k*Bell(k)*Bell(2*n-k), k = 0..2*n):
%p A000807 seq(A000807(n), n=0..17); # _Peter Luschny_, Sep 10 2017
%t A000807 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 A000807 (Python)
%o A000807 from sympy import binomial, bell
%o A000807 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 A000807 print([a(n) for n in range(21)]) # _Indranil Ghosh_, Sep 11 2017
%Y A000807 Cf. A000110.
%K A000807 nonn
%O A000807 0,2
%A A000807 _N. J. A. Sloane_
%E A000807 More terms from _Vladeta Jovovic_, Sep 08 2002