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A047819
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a(n) = Product_{i=1..n} ((i+3)*(i+4)*(i+5))/(i*(i+1)*(i+2)).
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16
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1, 20, 175, 980, 4116, 14112, 41580, 108900, 259545, 572572, 1184183, 2318680, 4331600, 7768320, 13441968, 22535064, 36729945, 58373700, 90684055, 138003404, 206108980, 302588000, 437287500, 622849500, 875343105, 1215006156, 1667110095
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OFFSET
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0,2
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COMMENTS
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Number of tilings of a <3,n,3> hexagon.
Determinant of the 3 X 3 matrix with rows [C(n+3,3) C(n+3,4) C(n+3,5)], [C(n+4,3) C(n+4,4) C(n+4,5)], and [C(n+5,3) C(n+5,4) C(n+5,5)]. - J. M. Bergot, Sep 10 2013
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REFERENCES
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O. D. Anderson, Find the next sequence, J. Rec. Math., 8 (No. 4, 1975-1976), 241.
S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 232, # 2 and p. 105, eq.(ii), K(0a(2,5,n))).
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LINKS
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Paolo 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]."]
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
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FORMULA
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G.f.: (1 + 10*x + 20*x^2 + 10*x^3 + x^4) / (1 - x)^10. - Michael Somos, Nov 14 2002
a(n-3) = (1/24)*Sum_{1 <= x_1, x_2, x_3 <= n} (det V(x_1,x_2,x_3))^2 = (1/24)*Sum_ {1 <= i,j,k <= n} ((i-j)(i-k)(j-k))^2, where V(x_1,x_2,x_3) is the Vandermonde matrix of order 3. - Peter Bala, Sep 21 2007
Sum_{n>=0} 1/a(n) = 5195/2 - 2160*zeta(3).
Sum_{n>=0} (-1)^n/a(n) = 17205/2 - 9600*log(2) - 1620*zeta(3). (End)
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EXAMPLE
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G.f. = 1 + 20*x + 175*x^2 + 980*x^3 + 4116*x^4 + 14112*x^5 + 41580*x^6 + ...
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MAPLE
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a:=n->(n+1)*(n+2)^2*(n+3)^3*(n+4)^2*(n+5)/8640: seq(a(n), n=0..30); # Emeric Deutsch, Jun 18 2005
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MATHEMATICA
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a[n_] :=(n + 1)*(n + 2)^2*(n + 3)^3*(n + 4)^2*(n + 5)/8640;
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PROG
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(PARI) {a(n) = if( n<0, 0, binomial(n+5, 5) * binomial(n+4, 3) * (n+3) / 12)}; /* Michael Somos, Nov 14 2002 */
(PARI) {a(n) = my(s=sign(n+3)); n=abs(n+3)-3; -s/8 * polcoeff( charpoly( matrix(n+3, n+3, i, j, (i-j)^2)), n)}; /* Michael Somos, Nov 14 2002 */
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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