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A006902 a(n) = (2n)! * Sum_{k=0..n} (-1)^k * binomial(n,k) / (n+k)!.

%I A006902 M4003 #69 Oct 27 2023 19:45:04
%S A006902 1,1,5,47,641,11389,248749,6439075,192621953,6536413529,248040482741,
%T A006902 10407123510871,478360626529345,23903857657114837,1290205338991689821,
%U A006902 74803882225482661259,4636427218380366565889,305927317398343461908785,21410426012751471702223333
%N A006902 a(n) = (2n)! * Sum_{k=0..n} (-1)^k * binomial(n,k) / (n+k)!.
%C A006902 Number of words on {1,1,2,2,...,n,n} with longest increasing subsequence of length n. a(2) = 5: 1122, 1212, 1221, 2112, 2121. - _Alois P. Heinz_, Jan 18 2016
%D A006902 J. D. Horton and A. Kurn, Counting sequences with complete increasing subsequences, Congress. Numerantium, 33 (1981), 75-80.
%D A006902 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006902 Alois P. Heinz, <a href="/A006902/b006902.txt">Table of n, a(n) for n = 0..360</a> (terms n = 1..100 from T. D. Noe)
%H A006902 Yassine El Maazouz and Jim Pitman, <a href="https://arxiv.org/abs/2210.02027">The Bernoulli clock: probabilistic and combinatorial interpretations of the Bernoulli polynomials by circular convolution</a>, arXiv:2210.02027 [math.PR], 2022.
%F A006902 a(n) = n!*LaguerreL(n, n, 1). - _Vladeta Jovovic_, May 11 2003
%F A006902 (n-2)*a(n) - (n^3+n^2-7*n+4)*a(n-1) + 2*(2*n-3)*(n-1)^3*a(n-2) = 0. - _Vladeta Jovovic_, Jul 16 2004
%F A006902 a(n) ~ n^n*2^(2*n+1/2)/exp(n+1). - _Vaclav Kotesovec_, Jun 22 2013
%F A006902 a(n) = B_n(n*0!,(n-1)*1!, ..., 1*(n-1)!), where B_n(x1, ..., xn) is the n-th complete Bell polynomial. - _Max Alekseyev_, Jul 04 2015
%F A006902 a(n) = n!*binomial(2*n,n)*hypergeom([-n], [n+1], 1). - _Peter Luschny_, May 04 2017
%F A006902 a(n) = n!*Z(S_n; n, n-1, ..., 1) where Z(S_n) is the cycle index of the symmetric group of order n. - _Sean A. Irvine_, Nov 14 2017
%F A006902 a(n) = n! * [x^n] exp(-x/(1 - x))/(1 - x)^(n+1). - _Ilya Gutkovskiy_, Nov 21 2017
%F A006902 E.g.f.: exp(1-c(x))/sqrt(1-4*x), where c(x) = (1-sqrt(1-4*x))/(2*x) is the Catalan number generating function. - _Ira M. Gessel_, Jun 04 2021
%p A006902 a:= proc(n) option remember; `if`(n<3, [1$2, 5][n+1], (
%p A006902       (n^3+n^2-7*n+4)*a(n-1)-2*(2*n-3)*(n-1)^3*a(n-2))/(n-2))
%p A006902     end:
%p A006902 seq(a(n), n=0..20);  # _Alois P. Heinz_, Jan 15 2016
%t A006902 Table[(-1)^k HypergeometricU[-k, 1+k, 1], {k,0,20}] (* _Vladimir Reshetnikov_, Feb 16 2011 *)
%o A006902 (PARI) a(n)=round(hyperu(-n,n+1,1)*(-1)^n) \\ _Charles R Greathouse IV_, Dec 30 2014
%o A006902 (Magma) [Factorial(n)*Evaluate(LaguerrePolynomial(n, n), 1): n in [0..40]]; // _G. C. Greubel_, Aug 11 2022
%o A006902 (SageMath) [factorial(n)*gen_laguerre(n,n,1) for n in (0..40)] # _G. C. Greubel_, Aug 11 2022
%Y A006902 Row n=2 of A047909.
%Y A006902 Main diagonal of A267480.
%Y A006902 Cf. A082545.
%K A006902 nonn,easy
%O A006902 0,3
%A A006902 _Simon Plouffe_ and _N. J. A. Sloane_
%E A006902 a(0)=1 prepended by _Alois P. Heinz_, Jan 15 2016