A000279 Card matching: coefficients B[n,1] of t in the reduced hit polynomial A[n,n,n](t).

3, 24, 216, 1824, 15150, 124416, 1014888, 8241792, 66724398, 538990800, 4346692680, 35009591040, 281699380560, 2264868936960, 18198009147600, 146142982814208, 1173123636533454, 9413509300965936, 75513633110271264, 605598295606296000, 4855626127979443908, 38924245740546950784

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Number of permutations of 3 distinct letters (ABC) each with n copies such that one (1) fixed points. E.g., if AAAAABBBBBCCCCC n=3*5 letters permutations then one fixed points n5=15150. - 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

• Alois P. Heinz, Table of n, a(n) for n = 1..300
• Index entries for sequences related to card matching

Crossrefs

Cf. A000489, A000535.

Cf. A033581.

Programs

Mathematica

f[n_] := HypergeometricPFQ[{-n, -n, -n}, {1, 1}, -1]; a[n_] := n^2*(f[n]+4*f[n-1])/(n+1); Array[a, 20] (* Jean-François Alcover, Mar 11 2014, after Mark van Hoeij *)

PARI

A000279(n)=3*n*sum(k=0,n-1,binomial(n,k+1)*binomial(n,k)*binomial(n-1,k)) \\ M. F. Hasler, Sep 21 2015

Formula

a(n) = 3n * sum(C(n, k+1)*C(n, k)*C(n-1, k), k=0..n-1).

G.f.: x * (6*hypergeom([4/3, 5/3],[2],27*x^2/(1-2*x)^3)/(1-2*x)^3 - 3*hypergeom([2/3, 4/3],[1],27*x^2/(1-2*x)^3)/(1-2*x)^2). - Mark van Hoeij, Oct 23 2011

a(n) = n^2*(A000172(n)+4*A000172(n-1))/(n+1). - Mark van Hoeij, Oct 26 2011

a(n) ~ 8^n*sqrt(3)/Pi = 8^n*0.5513... - M. F. Hasler, Sep 21 2015

a(n) = 3n*A262407(n). - M. F. Hasler, Sep 23 2015

Extensions

More terms from Vladeta Jovovic, Apr 26 2000

More terms from Emeric Deutsch, Feb 19 2004

Three lines of data completed and more explicit definition by M. F. Hasler, Sep 21 2015