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.

Previous Showing 11-14 of 14 results.

A051946 Expansion of g.f.: (1+4*x)/(1-x)^7.

Original entry on oeis.org

1, 11, 56, 196, 546, 1302, 2772, 5412, 9867, 17017, 28028, 44408, 68068, 101388, 147288, 209304, 291669, 399399, 538384, 715484, 938630, 1216930, 1560780, 1981980, 2493855, 3111381, 3851316, 4732336, 5775176, 7002776, 8440432
Offset: 0

Views

Author

Barry E. Williams, Dec 20 1999

Keywords

Comments

Kekulé numbers for certain benzenoids. - Emeric Deutsch, Jun 18 2005
Equals row sums of triangle A143130, and binomial transform of {1, 10, 35, 60, 55, 26, 5, 0, 0, 0, ...}. - Gary W. Adamson, Jul 27 2008

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
  • S. J. Cyvin and I. Gutman, KekulĂ© structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p.233, # 5).

Links

Crossrefs

Partial sums of A027800.
Cf. A093562 ((5, 1) Pascal, column m=6).
Cf. A143130.
Cf. similar sequences listed in A254142.

Programs

  • GAP
    List([0..40], n-> (5*n+6)*Binomial(n+5,5)/6); # G. C. Greubel, Aug 28 2019
  • Magma
    [(5*n+6)*Binomial(n+5,5)/6: n in [0..40]]; // Vincenzo Librandi, Jul 30 2014
  • Maple
    a:=n->(n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(5*n+6)/720: seq(a(n),n=0..35); # Emeric Deutsch
  • Mathematica
    CoefficientList[Series[(1+4x)/(1-x)^7, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 30 2014 *)
  • PARI
    vector(40, n, (5*n+1)*binomial(n+4,5)/6) \\ G. C. Greubel, Aug 28 2019
  • Sage
    [(5*n+6)*binomial(n+5,5)/6 for n in (0..40)] # G. C. Greubel, Aug 28 2019

Formula

a(n) = binomial(n+5,5)*(5*n+6)/6.
a(n) = (n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(5*n+6)/720. - Emeric Deutsch, Jun 18 2005
a(n) = A034264(n+1). - R. J. Mathar, Oct 14 2008

Extensions

Corrected and extended by Emeric Deutsch, Jun 18 2005

A050484 Partial sums of A051946.

Original entry on oeis.org

1, 12, 68, 264, 810, 2112, 4884, 10296, 20163, 37180, 65208, 109616, 177684, 279072, 426360, 635664, 927333, 1326732, 1865116, 2580600, 3519230, 4736160, 6296940, 8278920, 10772775, 13884156, 17735472, 22467808, 28242984, 35245760
Offset: 0

Views

Author

Barry E. Williams, Dec 26 1999

Keywords

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

Crossrefs

Cf. A051946, A093562 ((5, 1) Pascal, column m=7).

Programs

  • Magma
    [((5*n+7)*Binomial(n+6,6))/7: n in [0..60]]; // Vincenzo Librandi, Jul 30 2014
  • Mathematica
    CoefficientList[Series[(1 + 4 x)/(1 - x)^8, {x, 0, 50}], x] (* Vincenzo Librandi, Jul 30 2014 *)
Accumulate[CoefficientList[Series[(1+4x)/(1-x)^7,{x,0,40}],x]] (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,12,68,264,810,2112,4884,10296},30] (* Harvey P. Dale, Aug 21 2020 *)
  • PARI
    a(n)=binomial(n+6, 6)*(5*n+7)/7 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = C(n+6, 6)*(5n+7)/7.
G.f.: (1+4*x)/(1-x)^8.

A052255 Partial sums of A050484.

Original entry on oeis.org

1, 13, 81, 345, 1155, 3267, 8151, 18447, 38610, 75790, 140998, 250614, 428298, 707370, 1133730, 1769394, 2696727, 4023459, 5888575, 8469175, 11988405, 16724565, 23021505, 31300425
Offset: 0

Views

Author

Barry E. Williams, Feb 02 2000

Keywords

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
  • Murray R.Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pp. 10-20, 79-94.

Links

Crossrefs

Cf. A050484.
Cf. A093562 ((5, 1) Pascal, column m=8).

Formula

a(n) = (5n+8)*C(n+7, 7)/8.
G.f.: (1+4*x)/(1-x)^9.

A055844 a(n) = (5*n + 9)*binomial(n+8, 8)/9.

Original entry on oeis.org

1, 14, 95, 440, 1595, 4862, 13013, 31460, 70070, 145860, 286858, 537472, 965770, 1673140, 2806870, 4576264, 7272991, 11296450, 17185025, 25654200, 37642605, 54367170, 77388675, 108689100, 150762300, 206719656, 280412484, 376573120
Offset: 0

Views

Author

Barry E. Williams, May 30 2000

Keywords

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

Crossrefs

Cf. A052255.
Cf. A093562 ((5, 1) Pascal, column m=9).

Programs

  • GAP
    List([0..30], n-> (5*n+9)*Binomial(n+8, 8)/9); # G. C. Greubel, Jan 21 2020
  • Magma
    [(5*n+9)*Binomial(n+8, 8)/9: n in [0..30]]; // G. C. Greubel, Jan 21 2020
  • Maple
    seq((5*n+9)*binomial(n+8, 8)/9, n=0..30); # G. C. Greubel, Jan 21 2020
  • Mathematica
    Table[5*Binomial[n+9,9] -4*Binomial[n+8,8], {n,0,30}] (* G. C. Greubel, Jan 21 2020 *)
  • PARI
    vector(31, n, (5*n+4)*binomial(n+7, 8)/9) \\ G. C. Greubel, Jan 21 2020
  • Sage
    [(5*n+9)*binomial(n+8, 8)/9 for n in (0..30)] # G. C. Greubel, Jan 21 2020

Formula

a(n) = (5*n+9)*binomial(n+8, 8)/9.
G.f.: (1+4*x)/(1-x)^10.
From G. C. Greubel, Jan 21 2020: (Start)
a(n) = 5*binomial(n+9, 9) - 4*binomial(n+8, 8).
E.g.f.: (362880 +4717440*x +12337920*x^2 +11854080*x^3 +5292000*x^4 +1227744*x^5 +155232*x^6 +10656*x^7 +369*x^8 +5*x^9)*exp(x)/362880. (End)
Previous Showing 11-14 of 14 results.

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