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
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).
Programs
-
Magma
[(n+1)*StirlingSecond(n+3,n+1): n in [0..40]]; // 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 *) -
PARI
Vec((1 + 8*x + 6*x^2) / (1 - x)^6 + O(x^30)) \\ Colin Barker, Apr 22 2020
-
SageMath
[(n+1)*stirling_number2(n+3,n+1) for n in range(41)] # G. C. Greubel, Oct 19 2023
Formula
From Zerinvary Lajos, Jan 20 2007: (Start)
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
Comments