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 A114242 a(n) = (n+1)(n+2)^2*(n+3)^2*(n+4)(2n+5)/720. 6
 1, 14, 90, 385, 1274, 3528, 8568, 18810, 38115, 72358, 130130, 223587, 369460, 590240, 915552, 1383732, 2043621, 2956590, 4198810, 5863781, 8065134, 10939720, 14651000, 19392750, 25393095, 32918886, 42280434, 53836615, 68000360 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS KekulĂ© numbers for certain benzenoids. Partial sums of A114244. First differences of A006857. - Peter Bala, Sep 21 2007 REFERENCES S. J. Cyvin and I. Gutman, KekulĂ© structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (pp. 167-169, Table 10.5/II/2 and p. 105, eq. (ii) K(Ob(2,4,n))). LINKS Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1). FORMULA G.f.: (1+x)(1 + 5x + x^2)/(1-x)^8. a(n-2) = 1/6*sum_{1 <= x_1, x_2 <= n} (x_1)^2*x_2*(det V(x_1,x_2))^2 = 1/6*sum {1 <= i,j <= n} i^2*j*(i-j)^2, where V(x_1,x_2} is the Vandermonde matrix of order 2. - Peter Bala, Sep 21 2007 a(0)=1, a(1)=14, a(2)=90, a(3)=385, a(4)=1274, a(5)=3528, a(6)=8568, a(7)=18810, a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8). - Harvey P. Dale, Aug 21 2013 MAPLE a:=n->(n+1)*(n+2)^2*(n+3)^2*(n+4)*(2*n+5)/720: seq(a(n), n=0..30); MATHEMATICA Table[((n+1)(n+2)^2 (n+3)^2 (n+4)(2n+5))/720, {n, 0, 30}] (* or *) LinearRecurrence[ {8, -28, 56, -70, 56, -28, 8, -1}, {1, 14, 90, 385, 1274, 3528, 8568, 18810}, 30] (* Harvey P. Dale, Aug 21 2013 *) PROG (PARI) a(n)=(n+1)*(n+2)^2*(n+3)^2*(n+4)*(2*n+5)/720 \\ Charles R Greathouse IV, Oct 07 2015 CROSSREFS Cf. A005585, A006542, A107891. Sequence in context: A241305 A195267 A077538 * A054487 A200191 A266805 Adjacent sequences:  A114239 A114240 A114241 * A114243 A114244 A114245 KEYWORD nonn,easy AUTHOR Emeric Deutsch, Nov 18 2005 STATUS approved

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Last modified September 22 19:04 EDT 2018. Contains 315270 sequences. (Running on oeis4.)