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.

A047864 Number of labeled bipartite graphs with n nodes.

Original entry on oeis.org

1, 1, 2, 7, 41, 376, 5177, 103237, 2922446, 116011231, 6433447397, 498234407452, 54007795331921, 8213123246906761, 1756336596363006842, 528975889250504033527, 224688018516023267969441, 134708289561117007261966816
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 406.
  • H. S. Wilf, Generatingfunctionology, Academic Press, NY, 1990, p. 80, Eq. 3.11.5.

Links

Crossrefs

Row sums of A117279.
The unlabeled version is A033995.
Cf. A001832, A047863.

Programs

  • Mathematica
    nn = 20; a = Sum[Sum[Binomial[n, k] 2^(k (n - k)), {k, 0, n}] x^n/n!, {n, 0, nn}]; Range[0, nn]! CoefficientList[Series[a^(1/2), {x, 0, nn}], x]  (* Geoffrey Critzer, Jan 15 2012 *)
  • PARI
    N=18; x='x+O('x^N); Vec(serlaplace(sqrt(sum(n=0, N, exp(2^n*x)*x^n/n!)))) \\ Gheorghe Coserea, Nov 13 2017

Formula

E.g.f.: sqrt( e.g.f. for A047863 ).

A053526 Number of bipartite graphs with 3 edges on nodes {1..n}.

Original entry on oeis.org

0, 0, 0, 0, 16, 110, 435, 1295, 3220, 7056, 14070, 26070, 45540, 75790, 121121, 187005, 280280, 409360, 584460, 817836, 1124040, 1520190, 2026255, 2665355, 3464076, 4452800, 5666050, 7142850, 8927100, 11067966, 13620285
Offset: 0

Views

Author

N. J. A. Sloane, Jan 16 2000

Keywords

References

  • R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.5.

Links

Crossrefs

Column k=3 of A117279.
Cf. A000217 (1 edge), A050534 (2 edges).

Programs

  • GAP
    List([0..40], n-> Binomial(n,4)*(n^2+3*n+4)/2) # G. C. Greubel, May 15 2019
  • Magma
    [Binomial(n,4)*(n^2+3*n+4)/2: n in [0..40]]; // G. C. Greubel, May 15 2019
  • Mathematica
    Table[Binomial[n,4]*(n^2+3*n+4)/2, {n,0,40}] (* G. C. Greubel, May 15 2019 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,0,0,0,16,110,435},40] (* Harvey P. Dale, Nov 24 2022 *)
  • PARI
    {a(n) = binomial(n,4)*(n^2+3*n+4)/2}; \\ G. C. Greubel, May 15 2019
  • Sage
    [binomial(n,4)*(n^2+3*n+4)/2 for n in (0..40)] # G. C. Greubel, May 15 2019

Formula

a(n) = (n-3)*(n-2)*(n-1)*n*(n^2 + 3*n + 4)/48.
G.f.: x^4*(16-2*x+x^2)/(1-x)^7. - Colin Barker, May 08 2012
E.g.f.: x^4*(32 + 12*x + x^2)*exp(x)/48. - G. C. Greubel, May 15 2019

A053527 Number of bipartite graphs with 4 edges on nodes {1..n}.

Original entry on oeis.org

0, 0, 0, 0, 3, 140, 1125, 5355, 19075, 56133, 143955, 332475, 706860, 1404975, 2640638, 4733820, 8149050, 13543390, 21825450, 34227018, 52388985, 78463350, 115233195, 166252625, 236008773, 330108075, 455489125, 620664525, 835994250
Offset: 0

Views

Author

N. J. A. Sloane, Jan 16 2000

Keywords

References

  • R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.5.

Links

Crossrefs

Column k=4 of A117279.
Cf. A000217 (1 edge), A050534 (2 edges), A053526 (3 edges).

Programs

  • GAP
    List([0..30], n-> Binomial(n,4)*(n+2)*(n^3-5*n-36)/16 ) # G. C. Greubel, May 15 2019
  • Magma
    [(n^5-4*n^4-n^3+16*n^2-12*n)*(n^3-5*n-36)/384: n in [0..30]]; // Vincenzo Librandi, May 08 2012
  • Mathematica
    CoefficientList[Series[x^4*(3+113*x-27*x^2+18*x^3-2*x^4)/(1-x)^9, {x,0, 30}], x] (* Vincenzo Librandi, May 08 2012 *)
  • PARI
    {a(n) = binomial(n,4)*(n+2)*(n^3-5*n-36)/16}; \\ G. C. Greubel, May 15 2019
  • Sage
    [binomial(n,4)*(n+2)*(n^3-5*n-36)/16 for n in (0..30)] # G. C. Greubel, May 15 2019

Formula

a(n) = (n-3)*(n-2)*(n-1)*n*(n+2)*(n^3-5*n-36)/384.
G.f.: x^4*(3+113*x-27*x^2+18*x^3-2*x^4)/(1-x)^9. - Colin Barker, May 08 2012
E.g.f.: x^4*(48 + 400*x + 176*x^2 + 24*x^3 + x^4)*exp(x)/384. - G. C. Greubel, May 15 2019

A053528 Number of bipartite graphs with 5 edges on nodes {1..n}.

Original entry on oeis.org

0, 0, 0, 0, 0, 60, 1701, 14952, 81228, 331884, 1116675, 3256407, 8500734, 20306286, 45093048, 94189095, 186736368, 353904096, 644842674, 1134910242, 1936817820, 3215467584, 5207403663, 8245956642, 12793342716, 19481177100, 29161079805, 42967291185, 62393475690
Offset: 0

Views

Author

N. J. A. Sloane, Jan 16 2000

Keywords

References

  • R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.5.

Links

Crossrefs

Column k=5 of A117279.
Cf. A000217 (1 edge), A050534 (2 edges), A053526 (3 edges), A053527 (4 edges).

Programs

  • GAP
    List([0..30], n-> Binomial(n,5)*(n^5 +5*n^4 +5*n^3 -85*n^2 -374*n -960)/32) # G. C. Greubel, May 15 2019
  • Magma
    [Binomial(n,5)*(n^5 +5*n^4 +5*n^3 -85*n^2 -374*n -960)/32: n in [0..30]]; // G. C. Greubel, May 15 2019
  • Mathematica
    Table[Binomial[n,5]*(n^5 +5*n^4 +5*n^3 -85*n^2 -374*n -960)/32, {n,0,30}] (* G. C. Greubel, May 15 2019 *)
  • PARI
    {a(n) = binomial(n,5)*(n^5 +5*n^4 +5*n^3 -85*n^2 -374*n -960)/32}; \\ G. C. Greubel, May 15 2019
  • Sage
    [binomial(n,5)*(n^5 +5*n^4 +5*n^3 -85*n^2 -374*n -960)/32 for n in (0..30)] # G. C. Greubel, May 15 2019

Formula

a(n) = (n-4)*(n-3)*(n-2)*(n-1)*n*(n^5 + 5*n^4 + 5*n^3 - 85*n^2 - 374*n - 960)/3840.
G.f.: x^5*(60+1041*x-459*x^2+411*x^3-129*x^4+21*x^5)/(1-x)^11. - Colin Barker, May 08 2012
E.g.f.: x^5*(1920 + 7152*x + 3280*x^2 + 560*x^3 + 40*x^4 + x^5)*exp(x)/3840. - G. C. Greubel, May 15 2019
Showing 1-4 of 4 results.

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

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