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A000535
Card matching: coefficients B[n,2] of t^2 in the reduced hit polynomial A[n,n,n](t).
(Formerly M5194 N2258)
4
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).
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. A033581.
Sequence in context: A010979 A022591 A321954 * A251770 A033280 A125462
KEYWORD
nonn
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