A000279 Card matching: coefficients B[n,1] of t in the reduced hit polynomial A[n,n,n](t). (Formerly M3106 N1258)

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

Offset

1,1

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

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

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 *)

Prog

(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

Crossrefs

Cf. A000489, A000535.

Cf. A033581.

Sequence in context: A073978 A278991 A232692 * A370443 A292311 A370986

Adjacent sequences: A000276 A000277 A000278 * A000280 A000281 A000282

Keyword

nonn

Author

N. J. A. Sloane

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

Status

approved