A000535 - OEIS

%I M5194 N2258 #30 Feb 03 2022 02:30:45

%S 0,27,378,4536,48600,489780,4738104,44535456,409752432,3708359550,

%T 33125746500,292779558720,2565087894720,22307854940280,

%U 192788833482000,1657111548654720,14176605442521312,120779466450505758,1025230099571720676,8674221270307971600

%N Card matching: coefficients B[n,2] of t^2 in the reduced hit polynomial A[n,n,n](t).

%C Number of permutations of 3 distinct letters (ABC) each with n copies such that two (2) fixed points. E.g., if AAAAABBBBBCCCCC n=3*5 letters permutations then two fixed points n5=48600. - _Zerinvary Lajos_, Feb 02 2006

%C The definition uses notations of Riordan (1958), except for use of n instead of p. - _M. F. Hasler_, Sep 22 2015

%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 193.

%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 <a href="/index/Ca#cardmatch">Index entries for sequences related to card matching</a>

%F a(n) = 3*binomial(n, 2)*Sum_{k=0..n-2} binomial(n, k+2)*binomial(n, k)*binomial(n-2, k) + 3*n^2*Sum_{k=0..n-2} binomial(n, k+1)*binomial(n-1, k+1)*binomial(n-1, k).

%F a(n) = 3(n-1)*n^3 3F2(1-n, 1-n, 2-n; 2, 2; -1) + (3/4)(n-1)^2 n^2 3F2(2-n, 2-n, -n; 1, 3; -1), where 3F2 is the hypergeometric function 3F2. - _Jean-François Alcover_, Feb 09 2016

%F a(n) ~ 3^(3/2) * 2^(3*n - 2) * n / Pi. - _Vaclav Kotesovec_, Jun 10 2019

%t a[n_] := 3*Binomial[n, 2]*Sum[Binomial[n, k+2]*Binomial[n, k]*Binomial[n-2, k], {k, 0, n-2}] + 3n^2*Sum[Binomial[n, k+1]*Binomial[n-1, k+1]*Binomial[ n-1, k], {k, 0, n-2}] (* _Jean-François Alcover_, Feb 09 2016 *)

%o (PARI) A000535(n)=3*binomial(n,2)*sum(k=0,n-2,binomial(n,k+2)*binomial(n,k)*binomial(n-2,k))+3*n^2*sum(k=0,n-2,binomial(n,k+1)*binomial(n-1,k+1)*binomial(n-1,k)) \\ _M. F. Hasler_, Sep 30 2015

%Y Cf. A000279, A000489.

%Y Cf. A033581.

%K nonn

%O 1,2

%A _N. J. A. Sloane_

%E More terms from _Vladeta Jovovic_, Apr 26 2000

%E More terms from _Emeric Deutsch_, Feb 19 2004

%E More explicit definition by _M. F. Hasler_, Sep 22 2015