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URL: https://oeis.org/A053526

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A053526
Number of bipartite graphs with 3 edges on nodes {1..n}.
5
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
0,5
REFERENCES
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.5.
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
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 *)
PROG
(PARI) {a(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
(SageMath) [binomial(n, 4)*(n^2+3*n+4)/2 for n in (0..40)] # G. C. Greubel, May 15 2019
(GAP) List([0..40], n-> Binomial(n, 4)*(n^2+3*n+4)/2); # G. C. Greubel, May 15 2019
(Python)
def A053526(n): return n*(n*(n*(n*(n*(n - 3) - 3) + 3) + 26) - 24)//48 # Chai Wah Wu, Mar 17 2026
CROSSREFS
Column k=3 of A117279.
Cf. A000217 (1 edge), A050534 (2 edges).
Sequence in context: A238171 A155871 A120668 * A107908 A177046 A234250
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jan 16 2000
STATUS
approved