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

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

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

• Vincenzo Librandi, 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).

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