cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A108645 a(n) = (n+1)*(n+2)^2*(n+3)^2*(n+4)*(2*n^2 + 6*n + 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

Views

Author

Emeric Deutsch, Jun 13 2005

Keywords

Comments

Kekulé numbers for certain benzenoids.

References

  • 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).

Links

Crossrefs

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

Programs

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

Formula

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)*(11*n^3 + 58*n^2 + 101*n + 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

Views

Author

Emeric Deutsch, Jun 13 2005

Keywords

Comments

Kekulé numbers for certain benzenoids.

References

  • 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).

Links

Crossrefs

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

Programs

  • Magma
    [(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
  • Maple
    a:=(n+1)*(n+2)^2*(n+3)*(11*n^3+58*n^2+101*n+60)/720: seq(a(n),n=0..30);
  • Mathematica
    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 *)
  • Python
    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
  • SageMath
    [(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

Formula

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

A108649 a(n) = (n+1)*(n+2)*(n+3)*(13*n^3 + 69*n^2 + 113*n + 60)/360.

Original entry on oeis.org

1, 17, 111, 457, 1428, 3710, 8442, 17382, 33099, 59191, 100529, 163527, 256438, 389676, 576164, 831708, 1175397, 1630029, 2222563, 2984597, 3952872, 5169802, 6684030, 8551010, 10833615, 13602771, 16938117, 20928691, 25673642, 31282968
Offset: 0

Views

Author

Emeric Deutsch, Jun 13 2005

Keywords

Comments

Kekulé numbers for certain benzenoids.

Links

Crossrefs

Cf. A108645, A108646, A108647, A108648, A108650.

Programs

  • Magma
    [(13*n^3+69*n^2+113*n+60)*Binomial(n+3,3)/60: n in [0..40]]; // G. C. Greubel, Oct 19 2023
  • Maple
    a:=(n+1)*(n+2)*(n+3)*(13*n^3+69*n^2+113*n+60)/360: seq(a(n),n=0..36);
  • Mathematica
    Table[(n+1)(n+2)(n+3)(13n^3+69n^2+113n+60)/360,{n,0,30}] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1}, {1,17,111,457,1428,3710, 8442},30] (* Harvey P. Dale, Jul 01 2012 *)
  • PARI
    Vec((1+10*x+13*x^2+2*x^3)/(1-x)^7 + O(x^40)) \\ Colin Barker, Apr 22 2020
  • SageMath
    [(13*n^3+69*n^2+113*n+60)*binomial(n+3,3)/60 for n in range(41)] # G. C. Greubel, Oct 19 2023

Formula

a(0)=1, a(1)=17, a(2)=111, a(3)=457, a(4)=1428, a(5)=3710, a(6)=8442, a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7). - Harvey P. Dale, Jul 01 2012
G.f.: (1 + 10*x + 13*x^2 + 2*x^3) / (1 - x)^7. - Colin Barker, Apr 22 2020
E.g.f.: (1/360)*(360 + 5760*x + 14040*x^2 + 10440*x^3 + 2985*x^4 + 342*x^5 + 13*x^6)*exp(x). - G. C. Greubel, Oct 19 2023

A259455 n Sum_n Sum_n Sum_n.

Original entry on oeis.org

1, 30, 270, 1400, 5250, 15876, 41160, 95040, 200475, 393250, 726726, 1277640, 2153060, 3498600, 5508000, 8434176, 12601845, 18421830, 26407150, 37191000, 51546726, 70409900, 94902600, 126360000, 166359375, 216751626, 279695430, 357694120, 453635400, 570834000
Offset: 1

Views

Author

N. J. A. Sloane, Jun 30 2015

Keywords

Comments

See the reference for an explanation of the rather cryptic definition.

Links

Crossrefs

This is the seventh sequence in the sequence A000027, A000217, A002411, A001296, A108650, A001297, ...

Programs

  • Maple
    a:= n-> n^3*(n+3)*(n+2)*(n+1)^2/48:
seq(a(n), n=1..40);  # Alois P. Heinz, Jul 04 2015

Formula

From Alois P. Heinz, Jul 04 2015: (Start)
G.f.: (24*x^3+58*x^2+22*x+1)*x/(x-1)^8.
a(n) = n^3*(n+3)*(n+2)*(n+1)^2/48.
a(n) = n*Stirling2(n+3,n). (End)

Extensions

More terms from Alois P. Heinz, Jul 04 2015
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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