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%I #71 Oct 21 2022 21:59:10
%S 1,20,175,980,4116,14112,41580,108900,259545,572572,1184183,2318680,
%T 4331600,7768320,13441968,22535064,36729945,58373700,90684055,
%U 138003404,206108980,302588000,437287500,622849500,875343105,1215006156,1667110095
%N a(n) = Product_{i=1..n} ((i+3)*(i+4)*(i+5))/(i*(i+1)*(i+2)).
%C Number of tilings of a <3,n,3> hexagon.
%C Kekulé numbers for certain benzenoids. - _Emeric Deutsch_, Jun 18 2005
%C Partial sums of A107891. - _Peter Bala_, Sep 21 2007
%C 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
%D O. D. Anderson, Find the next sequence, J. Rec. Math., 8 (No. 4, 1975-1976), 241.
%D 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))).
%H Seiichi Manyama, <a href="/A047819/b047819.txt">Table of n, a(n) for n = 0..10000</a>
%H Paolo Aluffi, <a href="http://arxiv.org/abs/1408.1702">Degrees of projections of rank loci</a>, 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]."]
%H O. D. Anderson, <a href="/A002415/a002415.pdf">Find the next sequence</a>, J. Rec. Math., 8 (No. 4, 1975-1976), 241. [Annotated scanned copy]
%H Harald Helfgott and Ira M. Gessel, <a href="http://arXiv.org/abs/math.CO/9810143">Enumeration of tilings of diamonds and hexagons with defects</a>, arXiv:math/9810143 [math.CO], 1998.
%H J. M. Landsberg and L. Manivel, <a href="http://dx.doi.org/10.1016/j.aim.2005.02.001">The sextonions and E7 1/2</a>, Adv. Math. 201 (2006), 143-179. [Th. 7.2(ii), case a=2].
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
%F G.f.: (1 + 10*x + 20*x^2 + 10*x^3 + x^4) / (1 - x)^10. - _Michael Somos_, Nov 14 2002
%F a(n) = C(n+3,n+2)*C(n+4,n+1)*C(n+5,n)/12. - _Zerinvary Lajos_, May 29 2007
%F 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
%F a(n) = -a(-6-n) for all n in Z. - _Michael Somos_, Dec 26 2016
%F From _Amiram Eldar_, May 29 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 5195/2 - 2160*zeta(3).
%F Sum_{n>=0} (-1)^n/a(n) = 17205/2 - 9600*log(2) - 1620*zeta(3). (End)
%e G.f. = 1 + 20*x + 175*x^2 + 980*x^3 + 4116*x^4 + 14112*x^5 + 41580*x^6 + ...
%p 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
%t a[n_] :=(n + 1)*(n + 2)^2*(n + 3)^3*(n + 4)^2*(n + 5)/8640;
%t Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jul 19 2018, after _Emeric Deutsch_ *)
%o (PARI) {a(n) = if( n<0, 0, binomial(n+5, 5) * binomial(n+4, 3) * (n+3) / 12)}; /* _Michael Somos_, Nov 14 2002 */
%o (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 */
%Y Third row of array A103905.
%Y Cf. A002415, A107891, A107915, A133708.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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