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 A134287 Fifth column of triangle A103371 (without leading zeros). 2
 1, 30, 315, 1960, 8820, 31752, 97020, 261360, 637065, 1431430, 3006003, 5962320, 11262160, 20391840, 35581680, 60093504, 98590905, 157608990, 246142435, 376372920, 564559380, 832117000, 1206913500, 1724814000, 2431508625 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Kekulé numbers for certain benzenoids. a(n) = K(L(n))*K(O(2,4,n)) with the Cyvin and Gutman Kekulé number notation. See p. 62 for the L(n) structure with K(L(n))=n+1 and p. 105 (i) for the O(k,m,n) structure and its Kekulé number. This corresponds to an essentially disconnected 7-tier benzenoid structure similar to the 6-tier structure shown on p. 230, nr. 23 (see A108647). a(n-5), n >= 5, is the number of ways to put n identical objects into m=5 of altogether n distinguishable boxes (n-5 boxes stay empty). REFERENCES S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988. LINKS Reinhard Zumkeller, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1). FORMULA a(n) = A103371(n+4,4), n >= 0. a(n) = ((n+1)*(n+2)*(n+3)*(n+4))^2*(n+5)/2880, n >= 0. 2880 = 4!*5! = A010790(4). G.f.: (1+20*x+60*x^2+40*x^3+5*x^4)/(1-x)^10. Numerator polynomial from fifth row of triangle A132813. a(n) = 5*C(n+5,5)^2/(n+5), n >= 0. - Zerinvary Lajos, May 09 2008 a(n) = (C(n+6,6)*C(n+5,4)+5*C(n+5,6)*C(n+5,4))/(n+5). - Gary Detlefs, Jan 06 2014 From Amiram Eldar, May 31 2022: (Start) Sum_{n>=0} 1/a(n) = 350*Pi^2/3 - 13805/12. Sum_{n>=0} (-1)^n/a(n) = 5*Pi^2 + 640*log(2)/3 - 785/4. (End) E.g.f.: (2880 + 83520*x + 368640*x^2 + 529920*x^3 + 330120*x^4 + 102024*x^5 + 16616*x^6 + 1432*x^7 + 61*x^8 + x^9)*exp(x)/2880. - G. C. Greubel, Oct 28 2022 EXAMPLE a(2)=315 because n=7 identical balls can be put into m=5 of n=7 distinguishable boxes in binomial(7,5)*(5!/(4!*1!)+ 5!/(3!*2!)) = 21*(5+10) = 315 ways. The m=5 part partitions of 7, namely (1^4,3) and (1^3,2^2) specify the filling of each of the 21 possible five box choices. - Wolfdieter Lang, Nov 13 2007 MAPLE seq(binomial(n+4, 4)^2*(n+5)/5, n=0..24); # Peter Luschny, Jan 13 2014 MATHEMATICA CoefficientList[Series[(1 + 20 x + 60 x^2 + 40 x^3 + 5 x^4)/(1 - x)^10, {x, 0, 24}], x] PROG (MuPAD) 5*binomial(n+5, 5)^2/(n+5) \$ n = 0..35; // Zerinvary Lajos, May 09 2008 (PARI) a(n) = 5*binomial(n+5, 5)^2/(n+5); \\ Michel Marcus, Jan 07 2014 (Haskell) a134287 = flip a103371 4 . (+ 4) -- Reinhard Zumkeller, Apr 04 2014 (Magma) [5*Binomial(n+5, 5)^2/(n+5): n in [0..30]]; // G. C. Greubel, Oct 28 2022 (SageMath) [5*binomial(n+5, 5)^2/(n+5) for n in range(31)] # G. C. Greubel, Oct 28 2022 CROSSREFS Cf. A108647 (fourth column of triangle A103371). Sequence in context: A328751 A042750 A074994 * A141216 A159543 A227689 Adjacent sequences: A134284 A134285 A134286 * A134288 A134289 A134290 KEYWORD nonn,easy AUTHOR Wolfdieter Lang, Nov 13 2007 STATUS approved

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