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A006902 a(n) = (2n)! * Sum_{k=0..n} (-1)^k * binomial(n,k) / (n+k)!.
(Formerly M4003)
36
1, 1, 5, 47, 641, 11389, 248749, 6439075, 192621953, 6536413529, 248040482741, 10407123510871, 478360626529345, 23903857657114837, 1290205338991689821, 74803882225482661259, 4636427218380366565889, 305927317398343461908785, 21410426012751471702223333 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

REFERENCES

J. D. Horton and A. Kurn, Counting sequences with complete increasing subsequences, Congress. Numerantium, 33 (1981), 75-80.

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

LINKS

T. D. Noe and Alois P. Heinz, Table of n, a(n) for n = 0..360 (terms n = 1..100 from T. D. Noe)

FORMULA

a(n) = n!*LaguerreL(n, n, 1). - Vladeta Jovovic, May 11 2003

(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

a(n) ~ n^n*2^(2*n+1/2)/exp(n+1). - Vaclav Kotesovec, Jun 22 2013

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

a(n) = n!*binomial(2*n,n)*hypergeom([-n], [n+1], 1). - Peter Luschny, May 04 2017

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

a(n) = n! * [x^n] exp(-x/(1 - x))/(1 - x)^(n+1). - Ilya Gutkovskiy, Nov 21 2017

MAPLE

a:= proc(n) option remember; `if`(n<3, [1$2, 5][n+1], (

      (n^3+n^2-7*n+4)*a(n-1)-2*(2*n-3)*(n-1)^3*a(n-2))/(n-2))

    end:

seq(a(n), n=0..20);  # Alois P. Heinz, Jan 15 2016

MATHEMATICA

Table[(-1)^k HypergeometricU[-k, 1 + k, 1], {k, 1, 10}] (* Vladimir Reshetnikov, Feb 16 2011 *)

PROG

(PARI) a(n)=round(hyperu(-n, n+1, 1)*(-1)^n) \\ Charles R Greathouse IV, Dec 30 2014

CROSSREFS

Row n=2 of A047909.

Main diagonal of A267480.

Cf. A082545.

Sequence in context: A074192 A058806 A302616 * A180254 A127696 A088691

Adjacent sequences:  A006899 A006900 A006901 * A006903 A006904 A006905

KEYWORD

nonn,easy

AUTHOR

Simon Plouffe and N. J. A. Sloane.

EXTENSIONS

a(0)=1 prepended by Alois P. Heinz, Jan 15 2016

STATUS

approved

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Last modified June 16 19:43 EDT 2019. Contains 324155 sequences. (Running on oeis4.)