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A006542
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a(n) = binomial(n,3)*binomial(n-1,3)/4.
(Formerly M4707)
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25
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1, 10, 50, 175, 490, 1176, 2520, 4950, 9075, 15730, 26026, 41405, 63700, 95200, 138720, 197676, 276165, 379050, 512050, 681835, 896126, 1163800, 1495000, 1901250, 2395575, 2992626, 3708810, 4562425, 5573800, 6765440, 8162176, 9791320, 11682825, 13869450
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OFFSET
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4,2
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COMMENTS
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Number of permutations of n+4 that avoid the pattern 132 and have exactly 3 descents. - Mike Zabrocki, Aug 26 2004
a(n) = number of Dyck n-paths with exactly 4 peaks. - David Callan, Jul 03 2006
Six-dimensional figurate numbers for a hyperpyramid with pentagonal base. This corresponds to the sum(sum(sum(sum(1+sum(5*n))))) interpretation, see the Munafo webpage. - Robert Munafo, Jun 18 2009
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REFERENCES
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S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 166, no. 1).
S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 238.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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P. Aluffi, Degrees of projections of rank loci, arXiv:1408.1702 [math.AG], 2014. ["After compiling the results of many explicit computations, we noticed that many of the numbers d_{n,r,S} appear in the existing literature in contexts far removed from the enumerative geometry of rank conditions; we owe this surprising (to us) observation to perusal of [Slo14]."]
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FORMULA
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a(n) = C(n, 3)*C(n-1, 3)/4 = n*(n-1)^2*(n-2)^2*(n-3)/144.
a(n) = Sum(Sum(Sum(Sum(1 + Sum(5*n))))) = Sum (A006414). - Xavier Acloque, Oct 08 2003
a(n) = C(n, 6) + 3*C(n+1, 6) + C(n+2, 6). - Mike Zabrocki, Aug 26 2004
a(n) = C(n-2, n-4)*C(n-1, n-3)*C(n, n-2)/18. - Zerinvary Lajos, Jul 29 2005
a(n+2) = (1/4)*Sum_{1 <= x_1, x_2 <= n} x_1*x_2*(det V(x_1,x_2))^2 = (1/4)*Sum_{1 <= i,j <= n} i*j*(i-j)^2, where V(x_1,x_2} is the Vandermonde matrix of order 2. - Peter Bala, Sep 21 2007
a(n) = C(n-1,3)^2 - C(n-1,2)*C(n-1,4). - Gary Detlefs, Dec 05 2011
Sum_{n>=4} (-1)^n/a(n) = 134 - 192*log(2). - Amiram Eldar, Oct 19 2020
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MAPLE
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MATHEMATICA
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Table[Binomial[n, 3]*Binomial[n-1, 3]/4, {n, 4, 40}]
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PROG
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(PARI) a(n)=n*((n-1)*(n-2))^2*(n-3)/144
(Magma) [ n*((n-1)*(n-2))^2*(n-3)/144 : n in [4..40] ]; // Wesley Ivan Hurt, Jun 17 2014
(Sage) [n*(n-1)^2*(n-2)^2*(n-3)/144 for n in (4..40)] # G. C. Greubel, Feb 24 2019
(GAP) List([4..40], n-> n*(n-1)^2*(n-2)^2*(n-3)/144) # G. C. Greubel, Feb 24 2019
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CROSSREFS
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The expression binomial(m+n-1,n)^2-binomial(m+n,n+1)*binomial(m+n-2,n-1) for the values m = 2 through 14 produces the sequences A000012, A000217, A002415, A006542, A006857, A108679, A134288, A134289, A134290, A134291, A140925, A140935, A169937.
Fourth column of the table of Narayana numbers A001263.
Apart from a scale factor, a column of A124428.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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Zabroki and Lajos formulas offset corrected by Gary Detlefs, Dec 05 2011
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STATUS
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approved
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