A000535 Card matching: coefficients B[n,2] of t^2 in the reduced hit polynomial A[n,n,n](t). (Formerly M5194 N2258)

0, 27, 378, 4536, 48600, 489780, 4738104, 44535456, 409752432, 3708359550, 33125746500, 292779558720, 2565087894720, 22307854940280, 192788833482000, 1657111548654720, 14176605442521312, 120779466450505758, 1025230099571720676, 8674221270307971600

Offset

1,2

Comments

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

The definition uses notations of Riordan (1958), except for use of n instead of p. - M. F. Hasler, Sep 22 2015

References

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

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

Links

Table of n, a(n) for n=1..20.

Index entries for sequences related to card matching

Formula

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).

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

a(n) ~ 3^(3/2) * 2^(3*n - 2) * n / Pi. - Vaclav Kotesovec, Jun 10 2019

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A000279, A000489.

Cf. A033581.

Sequence in context: A010979 A022591 A321954 * A251770 A033280 A125462

Adjacent sequences: A000532 A000533 A000534 * A000536 A000537 A000538

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Apr 26 2000

More terms from Emeric Deutsch, Feb 19 2004

More explicit definition by M. F. Hasler, Sep 22 2015

Status

approved