A108649 -id:A108649 - cp's OEIS frontend

A108650 a(n) = (n+1)^2(n+2)(n+3)(3n+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

Emeric Deutsch, Jun 13 2005

Kekulé numbers for certain benzenoids.

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).

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

Cf. A001296, A001477.

[(n+1)*StirlingSecond(n+3,n+1): n in [0..40]]; // G. C. Greubel, Oct 19 2023

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

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 *)

Vec((1 + 8*x + 6*x^2) / (1 - x)^6 + O(x^30)) \\ Colin Barker, Apr 22 2020

[(n+1)*stirling_number2(n+3,n+1) for n in range(41)] # G. C. Greubel, Oct 19 2023

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

A108645 a(n) = (n+1)(n+2)^2(n+3)^2(n+4)(2n^2 + 6n + 5)/720.

Original entry on oeis.org

1, 26, 250, 1435, 5978, 19992, 56952, 143550, 328515, 695266, 1379378, 2591953, 4650100, 8015840, 13344864, 21546684, 33857829, 51929850, 77934010, 114684647, 165783310, 235785880, 330395000, 456680250, 623328615, 840927906, 1122285906

Offset: 0

Emeric Deutsch, Jun 13 2005

nonn

Kekulé numbers for certain benzenoids.

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 230, no. 21).

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A108646, A108647, A108648, A108649, A108650.

B:=Binomial; [(2*n^2+6*n+5)*B(n+4,4)*B(n+3,2)/15: n in [0..40]]; // G. C. Greubel, Oct 19 2023

a:=(n+1)*(n+2)^2*(n+3)^2*(n+4)*(2*n^2+6*n+5)/720: seq(a(n),n=0..30);

Table[(n+1)(n+2)^2(n+3)^2(n+4)(2n^2+6n+5)/720,{n,0,30}] (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,26,250,1435, 5978,19992,56952,143550,328515},30] (* Harvey P. Dale, Sep 05 2016 *)

b=binomial; [(2*n^2+6*n+5)*b(n+4,4)*b(n+3,2)/15 for n in range(41)] # G. C. Greubel, Oct 19 2023

G.f.: (1+17*x+52*x^2+37*x^3+5*x^4)/(1-x)^9. - Harvey P. Dale, Sep 05 2016

E.g.f.: (1/6!)*(720 + 18000*x + 71640*x^2 + 91440*x^3 + 49050*x^4 + 12486*x^5 + 1565*x^6 + 92*x^7 + 2*x^8)*exp(x). - G. C. Greubel, Oct 19 2023

A108646 a(n) = (n+1)(n+2)^2(n+3)(11n^3 + 58n^2 + 101n + 60)/720.

Original entry on oeis.org

1, 23, 194, 985, 3668, 11074, 28728, 66438, 140415, 276001, 511082, 900263, 1519882, 2473940, 3901024, 5982300, 8950653, 13101051, 18802210, 26509637, 36780128, 50287798, 67841720, 90405250, 119117115, 155314341, 200557098

Offset: 0

Emeric Deutsch, Jun 13 2005

Kekulé numbers for certain benzenoids.

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 230, no. 22).

Chai Wah Wu, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A108645, A108647, A108648, A108649, A108650.

[(n+2)*(11*n^3+58*n^2+101*n+60)*Binomial(n+3,3)/120: n in [0..40]]; // G. C. Greubel, Oct 19 2023

a:=(n+1)*(n+2)^2*(n+3)*(11*n^3+58*n^2+101*n+60)/720: seq(a(n),n=0..30);

Table[(n+2)*(n+3)!*(11*n^3+58*n^2+101*n+60)/(6!*n!), {n,0,40}] (* G. C. Greubel, Oct 19 2023 *)

A108646_list, m = [], [77, -85, 28, -1, 1, 1, 1, 1]
for _ in range(10001):
    A108646_list.append(m[-1])
    for i in range(7):
        m[i+1] += m[i] # Chai Wah Wu, Jun 12 2016

[(n+2)*(11*n^3+58*n^2+101*n+60)*binomial(n+3,3)/120 for n in range(41)] # G. C. Greubel, Oct 19 2023

From Chai Wah Wu, Jun 12 2016: (Start)
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) for n > 7.
G.f.: (1 + 15*x + 38*x^2 + 21*x^3 + 2*x^4)/(1 - x)^8. (End)
E.g.f.: (1/6!)*(720 + 15840*x + 53640*x^2 + 56520*x^3 + 24030*x^4 + 4548*x^5 + 377*x^6 + 11*x^7)*exp(x). - G. C. Greubel, Oct 19 2023

cp's OEIS Frontend

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

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A108645 a(n) = (n+1)*(n+2)^2*(n+3)^2*(n+4)*(2*n^2 + 6*n + 5)/720.

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Author

Keywords

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Links

Crossrefs

Programs

Magma

Maple

Mathematica

SageMath

Formula

A108646 a(n) = (n+1)*(n+2)^2*(n+3)*(11*n^3 + 58*n^2 + 101*n + 60)/720.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Python

SageMath

Formula

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

A108645 a(n) = (n+1)(n+2)^2(n+3)^2(n+4)(2n^2 + 6n + 5)/720.

A108646 a(n) = (n+1)(n+2)^2(n+3)(11n^3 + 58n^2 + 101n + 60)/720.