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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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Last modified November 30 01:31 EST 2022. Contains 358431 sequences. (Running on oeis4.)