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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
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
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 A362521
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Nov 13 2007
STATUS
approved