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 A108647 a(n) = (n+1)^2*(n+2)^2*(n+3)^2*(n+4)/144. 4
 1, 20, 150, 700, 2450, 7056, 17640, 39600, 81675, 157300, 286286, 496860, 828100, 1332800, 2080800, 3162816, 4694805, 6822900, 9728950, 13636700, 18818646, 25603600, 34385000, 45630000, 59889375, 77808276, 100137870, 127747900 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS KekulĂ© numbers for certain benzenoids. a(n-4), n>=4, is the number of ways to have n identical objects in m=4 of altogether n distinguishable boxes (n-4 boxes stay empty). - Wolfdieter Lang, Nov 13 2007 REFERENCES S. J. Cyvin and I. Gutman, KekulĂ© structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 230, no. 23). LINKS Reinhard Zumkeller, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1). FORMULA a(n) = C(n+4,4)*C(n+3,2)(n+1)/3. - Paul Barry, May 13 2006 G.f.: (1+12*x+18*x^2+4*x^3)/(1-x)^8. a(n) = 4*C(n,4)^2/n, n >= 4. - Zerinvary Lajos, May 09 2008 EXAMPLE a(2)=150 because n=6 identical balls can be put into m=4 of n=6 distinguishable boxes in binomial(6,4)*(4!/(3!*1!)+ 4!/(2!*2!)) = 15*(4 + 6) =150 ways. The m=4 part partitions of 6, namely (1^3,3) and (1^2,2^2) specify the filling of each of the 15 possible four box choices. - Wolfdieter Lang, Nov 13 2007 MAPLE a:=(n+1)^2*(n+2)^2*(n+3)^2*(n+4)/144: seq(a(n), n=0..30); MATHEMATICA Array[Binomial[# + 4, 4] Binomial[# + 3, 2] (# + 1)/3 &, 28, 0] (* or *) CoefficientList[Series[(1 + 12 x + 18 x^2 + 4 x^3)/(1 - x)^8, {x, 0, 27}], x] (* Michael De Vlieger, Dec 17 2017 *) PROG (MuPAD) 4*binomial(n, 4)^2/n \$ n = 4..35; // Zerinvary Lajos, May 09 2008 (Haskell) a108647 = flip a103371 3 . (+ 3)  -- Reinhard Zumkeller, Apr 04 2014 CROSSREFS Fourth column of triangle A103371. Sequence in context: A100190 A189494 A022680 * A164605 A000492 A015866 Adjacent sequences:  A108644 A108645 A108646 * A108648 A108649 A108650 KEYWORD nonn AUTHOR Emeric Deutsch, Jun 13 2005 STATUS approved

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Last modified December 2 07:25 EST 2020. Contains 338868 sequences. (Running on oeis4.)