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

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

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

• Index entries for sequences related to card matching

Crossrefs

Cf. A000279, A000489.

Cf. A033581.

Programs

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 *)

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

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

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