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%I #27 Nov 24 2022 12:41:19
%S 0,0,0,0,16,110,435,1295,3220,7056,14070,26070,45540,75790,121121,
%T 187005,280280,409360,584460,817836,1124040,1520190,2026255,2665355,
%U 3464076,4452800,5666050,7142850,8927100,11067966,13620285
%N Number of bipartite graphs with 3 edges on nodes {1..n}.
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.5.
%H G. C. Greubel, <a href="/A053526/b053526.txt">Table of n, a(n) for n = 0..1000</a>
%H Chai Wah Wu, <a href="http://arxiv.org/abs/1407.5663">Graphs whose normalized Laplacian matrices are separable as density matrices in quantum mechanics</a>, arXiv:1407.5663 [quant-ph], 2014.
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F a(n) = (n-3)*(n-2)*(n-1)*n*(n^2 + 3*n + 4)/48.
%F G.f.: x^4*(16-2*x+x^2)/(1-x)^7. - _Colin Barker_, May 08 2012
%F E.g.f.: x^4*(32 + 12*x + x^2)*exp(x)/48. - _G. C. Greubel_, May 15 2019
%t Table[Binomial[n,4]*(n^2+3*n+4)/2, {n,0,40}] (* _G. C. Greubel_, May 15 2019 *)
%t LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,0,0,0,16,110,435},40] (* _Harvey P. Dale_, Nov 24 2022 *)
%o (PARI) {a(n) = binomial(n,4)*(n^2+3*n+4)/2}; \\ _G. C. Greubel_, May 15 2019
%o (Magma) [Binomial(n,4)*(n^2+3*n+4)/2: n in [0..40]]; // _G. C. Greubel_, May 15 2019
%o (Sage) [binomial(n,4)*(n^2+3*n+4)/2 for n in (0..40)] # _G. C. Greubel_, May 15 2019
%o (GAP) List([0..40], n-> Binomial(n,4)*(n^2+3*n+4)/2) # _G. C. Greubel_, May 15 2019
%Y Column k=3 of A117279.
%Y Cf. A000217 (1 edge), A050534 (2 edges).
%K nonn,easy
%O 0,5
%A _N. J. A. Sloane_, Jan 16 2000
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