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A003517
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Number of permutations of [n+1] with exactly 1 increasing subsequence of length 3.
(Formerly M4177)
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32
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1, 6, 27, 110, 429, 1638, 6188, 23256, 87210, 326876, 1225785, 4601610, 17298645, 65132550, 245642760, 927983760, 3511574910, 13309856820, 50528160150, 192113383644, 731508653106, 2789279908316, 10649977831752, 40715807302800
(list;
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refs;
listen;
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OFFSET
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2,2
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COMMENTS
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a(n-4) = number of n-th generation vertices in the tree of sequences with unit increase labeled by 5 (cf. Zoran Sunik reference) - Benoit Cloitre, Oct 07 2003
Number of standard tableaux of shape (n+3,n-2). - Emeric Deutsch, May 30 2004
a(n) = A214292(2*n,n-3) for n > 2. - Reinhard Zumkeller, Jul 12 2012
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REFERENCES
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S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395-405.
J. Noonan, The number of permutations containing exactly one increasing subsequence of length three, Discrete Math. 152 (1996), no. 1-3, 307-313.
L. W. Shapiro, A Catalan triangle, Discrete Math. 14 (1976), no. 1, 83-90.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Zoran Sunik, Self describing sequences and the Catalan family tree, Elect. J. Combin., 10 (No. 1, 2003).
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 2..1000
D. Callan, A recursive bijective approach to counting permutations...
R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6
J. Noonan and D. Zeilberger, [math/9808080] The Enumeration of Permutations With a Prescribed Number of ``Forbidden'' Patterns
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FORMULA
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a(n) = 6*C(2*n+1, n-2)/(n+4).
G.f.: x^2*C(x)^6, where C(x) is g.f. for the Catalan numbers (A000108). - Emeric Deutsch, May 30 2004
E.g.f.: exp(2*x)*(Bessel_I(2,2*x)-Bessel_I(4,2*x)); - Paul Barry, Jun 04 2007
Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=5, a(n-3)=(-1)^(n-5)*coeff(charpoly(A,x),x^5). - Milan Janjic, Jul 08 2010
a(n)=sum(Catalan(i)*Catalan(j)*Catalan(k), i>=1,j>=1,k>=1, i+j+k=n+1)
-(n+4)*(n-2)*a(n) +2*n*(2*n+1)*a(n-1)=0. - R. J. Mathar, Dec 04 2012
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EXAMPLE
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a(3)=6 because the only permutations of 1234 with exactly 1 increasing subsequence of length 3 are 1423, 4123, 1342, 2314, 2341, 3124.
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MATHEMATICA
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f[x_] = (Sqrt[1 - 4 x] - 1)^6/(64 x^4); CoefficientList[Series[f[x], {x, 0, 25}], x][[3 ;; 26]] (* Jean-François Alcover, Jul 13 2011, after g.f. *)
Table[6 Binomial[2n+1, n-2]/(n+4), {n, 2, 30}] (* Harvey P. Dale, Feb 27 2012 *)
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PROG
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(PARI) a(n)=6*binomial(2*n+1, n-2)/(n+4) \\ Charles R Greathouse IV, May 18 2015
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CROSSREFS
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T(n, n+6) for n=0, 1, 2, ..., array T as in A047072.
Cf. A001089, A084249, A000108, A000245, A002057, A000344, A000588, A003518, A003519, A001392.
First differences are in A026017.
A diagonal of any of the essentially equivalent arrays: A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
Sequence in context: A094788 A221863 A216263 * A108958 A005284 A198694
Adjacent sequences: A003514 A003515 A003516 * A003518 A003519 A003520
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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