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%I #55 Aug 15 2020 16:02:29
%S 1,1,4,24,204,2220,29520,463680,8401680,172504080,3958113600,
%T 100370793600,2787459998400,84139894238400,2742857884166400,
%U 96034297911552000,3594206259195552000,143193586818810528000,6050501147565883008000,270263264589232282368000
%N a(n+2) = (2n+3)*a(n+1) + (n+1)^2*a(n), a(0) = 1, a(1) = 1.
%C a(n) is the number of n-letter words from an n-letter alphabet such that no letter appears more than twice. - _Paul Boddington_, Nov 17 2003
%H Alois P. Heinz, <a href="/A012244/b012244.txt">Table of n, a(n) for n = 0..393</a>
%H K. S. Brown, <a href="https://www.mathpages.com/home/kmath381/kmath381.htm">Integer Sequences Related To Pi</a>
%H D. Dominici, <a href="http://arxiv.org/abs/math/0501052">Nested derivatives: A simple method for computing series expansions of inverse functions</a>, arXiv:math/0501052 [math.CA], 2005.
%F E.g.f.: A(x) = (1 - 2*x - x^2)^(-1/2). - _Paul Boddington_, Nov 17 2003
%F a(n) = n!/2^n*A006139(n) = n!*Sum_{k=floor(n/2)..n} 2^(k-n)*C(n, k)*C(k, n-k). Sum_{n>=0} a(n)*x^n/n!^2 = exp(x)*BesselI(0, sqrt(2)*x). a(n) is the central coefficient of n!*(1+x+x^2/2)^n. - _Vladeta Jovovic_, Mar 22 2004
%F From _Peter Bala_, Aug 25 2011: (Start)
%F The function B(x) := int {t=0..x} A(t), obtained by integrating the generating function A(x), satisfies the autonomous differential equation d/dx(B(x)) = 1/(cos(B(x))-sin(B(x))). Compare with A190392.
%F Thus B(x), and hence A(x), can be found by inverting the function int {t=0..x} (cos(t)-sin(t)). By applying [Dominici, Theorem 4.1] the result can be expressed as
%F A(x) = 1 + sum {n>=1} D^n[1/(cos(t)-sin(t))](0)*x^n/n!, where the nested derivative D^n[f](x) of a function f(x) is defined recursively as D^0[f](x) = 1 and D^(n+1)[f](x) = d/dx(f(x)*D^n[f](x)) for n >= 0. Thus a(n) = D^n[1/(cos(t)-sin(t))](0). (End)
%F E.g.f. at offset 1: Series_Reversion(cos(x) + sin(x) - 1). - _Paul D. Hanna_, Aug 08 2012
%F a(n) ~ (1+sqrt(2))^(n+1/2) * n^n / (2^(1/4) * exp(n)). - _Vaclav Kotesovec_, Feb 18 2017
%p f := proc(n) option remember; if n <= 1 then 1 else (2*n-1)*f(n-1) +(n-1)^2*f(n-2); fi; end;
%t Range[0,20]! CoefficientList[Series[1/(1-2x-x^2)^(1/2), {x,0,20}], x] (* _Geoffrey Critzer_, Dec 07 2011 *)
%o (PARI) {a(n)=local(X=x+x^2*O(x^n));(n+1)!*polcoeff(serreverse(cos(X)+sin(X)-1),n+1)} \\ _Paul D. Hanna_, Aug 08 2012
%Y Cf. A089975, A190392, A054765, A054766.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
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