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%I M3106 N1258 #46 Sep 23 2015 20:34:56
%S 3,24,216,1824,15150,124416,1014888,8241792,66724398,538990800,
%T 4346692680,35009591040,281699380560,2264868936960,18198009147600,
%U 146142982814208,1173123636533454,9413509300965936,75513633110271264,605598295606296000,4855626127979443908,38924245740546950784
%N Card matching: coefficients B[n,1] of t 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 one (1) fixed points. E.g., if AAAAABBBBBCCCCC n=3*5 letters permutations then one fixed points n5=15150. - _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 Alois P. Heinz, <a href="/A000279/b000279.txt">Table of n, a(n) for n = 1..300</a>
%H <a href="/index/Ca#cardmatch">Index entries for sequences related to card matching</a>
%F a(n) = 3n * sum(C(n, k+1)*C(n, k)*C(n-1, k), k=0..n-1).
%F 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
%F a(n) = n^2*(A000172(n)+4*A000172(n-1))/(n+1). - _Mark van Hoeij_, Oct 26 2011
%F a(n) ~ 8^n*sqrt(3)/Pi = 8^n*0.5513... - _M. F. Hasler_, Sep 21 2015
%F a(n) = 3n*A262407(n). - _M. F. Hasler_, Sep 23 2015
%t 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_ *)
%o (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
%Y Cf. A000489, A000535.
%Y Cf. A033581.
%K nonn
%O 1,1
%A _N. J. A. Sloane_
%E More terms from _Vladeta Jovovic_, Apr 26 2000
%E More terms from _Emeric Deutsch_, Feb 19 2004
%E Three lines of data completed and more explicit definition by _M. F. Hasler_, Sep 21 2015
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