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

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

Author

N. J. A. Sloane, Jan 16 2000

Keywords

• nonn
• easy

References

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Chai Wah Wu, Graphs whose normalized Laplacian matrices are separable as density matrices in quantum mechanics, arXiv:1407.5663 [quant-ph], 2014.
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

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