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%I #24 Sep 08 2022 08:45:00
%S 0,0,0,0,3,140,1125,5355,19075,56133,143955,332475,706860,1404975,
%T 2640638,4733820,8149050,13543390,21825450,34227018,52388985,78463350,
%U 115233195,166252625,236008773,330108075,455489125,620664525,835994250
%N Number of bipartite graphs with 4 edges on nodes {1..n}.
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.5.
%H Vincenzo Librandi, <a href="/A053527/b053527.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F a(n) = (n-3)*(n-2)*(n-1)*n*(n+2)*(n^3-5*n-36)/384.
%F G.f.: x^4*(3+113*x-27*x^2+18*x^3-2*x^4)/(1-x)^9. - _Colin Barker_, May 08 2012
%F 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
%t 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 *)
%o (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
%o (PARI) {a(n) = binomial(n,4)*(n+2)*(n^3-5*n-36)/16}; \\ _G. C. Greubel_, May 15 2019
%o (Sage) [binomial(n,4)*(n+2)*(n^3-5*n-36)/16 for n in (0..30)] # _G. C. Greubel_, May 15 2019
%o (GAP) List([0..30], n-> Binomial(n,4)*(n+2)*(n^3-5*n-36)/16 ) # _G. C. Greubel_, May 15 2019
%Y Column k=4 of A117279.
%Y Cf. A000217 (1 edge), A050534 (2 edges), A053526 (3 edges).
%K nonn,easy
%O 0,5
%A _N. J. A. Sloane_, Jan 16 2000
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