A108650 a(n) = (n+1)^2*(n+2)*(n+3)*(3*n+4)/24.

1, 14, 75, 260, 700, 1596, 3234, 6000, 10395, 17050, 26741, 40404, 59150, 84280, 117300, 159936, 214149, 282150, 366415, 469700, 595056, 745844, 925750, 1138800, 1389375, 1682226, 2022489, 2415700, 2867810, 3385200, 3974696, 4643584

Offset

0,2

Comments

Kekulé numbers for certain benzenoids.

Links

Bo Gyu Jeong, Table of n, a(n) for n = 0..5000

Somaya Barati, Beáta Bényi, Abbas Jafarzadeh, and Daniel Yaqubi, Mixed restricted Stirling numbers, arXiv:1812.02955 [math.CO], 2018.

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 230, no. 26).

C. Krishnamachari, The operator (xD)^n, J. Indian Math. Soc., 15 (1923),3-4. [Annotated scanned copy]

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Formula

From Zerinvary Lajos, Jan 20 2007: (Start)

a(n) = A001477(n+1)*A001296(n+1) = (n+1)*A001296(n+1).

a(n) = (n+1)*Stirling2(n+3,n+1). (End)

From Colin Barker, Apr 22 2020: (Start)

G.f.: (1 + 8*x + 6*x^2) / (1 - x)^6.

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>5.

(End)

From Amiram Eldar, May 29 2022: (Start)

Sum_{n>=0} 1/a(n) 2*Pi^2 + 54*sqrt(3)*Pi/5 + 486*log(3)/5 - 921/5.

Sum_{n>=0} (-1)^n/a(n) = Pi^2 - 108*sqrt(3)*Pi/5 - 528*log(2)/5 + 909/5. (End)

E.g.f.: (1/24)*(24 +312*x +576*x^2 +304*x^3 +55*x^4 +3*x^5)*exp(x). - G. C. Greubel, Oct 19 2023

Maple

a:= n-> (n+1)^2*(n+2)*(n+3)*(3*n+4)/24: seq(a(n), n=0..36);
seq((n+1)*stirling2(n+3, n+1), n=0..32); # Zerinvary Lajos, Jan 20 2007

Mathematica

Table[((n+1)^2 (n+2)(n+3)(3n+4))/24, {n, 0, 40}] (* or *) Table[n StirlingS2[n+2, n], {n, 40}] (* Harvey P. Dale, Dec 01 2013 *)

Prog

(PARI) Vec((1 + 8*x + 6*x^2) / (1 - x)^6 + O(x^30)) \\ Colin Barker, Apr 22 2020
(Magma) [(n+1)*StirlingSecond(n+3, n+1): n in [0..40]]; // G. C. Greubel, Oct 19 2023
(SageMath) [(n+1)*stirling_number2(n+3, n+1) for n in range(41)] # G. C. Greubel, Oct 19 2023

Crossrefs

Cf. A108645, A108646, A108647, A108648, A108649.

Cf. A001296, A001477.

Sequence in context: A205583 A167633 A196411 * A093567 A296996 A270704

Adjacent sequences: A108647 A108648 A108649 * A108651 A108652 A108653

Keyword

nonn,easy

Author

Emeric Deutsch, Jun 13 2005

Status

approved