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A053528 Number of bipartite graphs with 5 edges on nodes {1..n}.

%I A053528 #21 Sep 08 2022 08:45:00
%S A053528 0,0,0,0,0,60,1701,14952,81228,331884,1116675,3256407,8500734,
%T A053528 20306286,45093048,94189095,186736368,353904096,644842674,1134910242,
%U A053528 1936817820,3215467584,5207403663,8245956642,12793342716,19481177100,29161079805,42967291185,62393475690
%N A053528 Number of bipartite graphs with 5 edges on nodes {1..n}.
%D A053528 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.5.
%H A053528 T. D. Noe, <a href="/A053528/b053528.txt">Table of n, a(n) for n = 0..1000</a>
%H A053528 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
%F A053528 a(n) = (n-4)*(n-3)*(n-2)*(n-1)*n*(n^5 + 5*n^4 + 5*n^3 - 85*n^2 - 374*n - 960)/3840.
%F A053528 G.f.: x^5*(60+1041*x-459*x^2+411*x^3-129*x^4+21*x^5)/(1-x)^11. - _Colin Barker_, May 08 2012
%F A053528 E.g.f.: x^5*(1920 + 7152*x + 3280*x^2 + 560*x^3 + 40*x^4 + x^5)*exp(x)/3840. - _G. C. Greubel_, May 15 2019
%t A053528 Table[Binomial[n,5]*(n^5 +5*n^4 +5*n^3 -85*n^2 -374*n -960)/32, {n,0,30}] (* _G. C. Greubel_, May 15 2019 *)
%o A053528 (PARI) {a(n) = binomial(n,5)*(n^5 +5*n^4 +5*n^3 -85*n^2 -374*n -960)/32}; \\ _G. C. Greubel_, May 15 2019
%o A053528 (Magma) [Binomial(n,5)*(n^5 +5*n^4 +5*n^3 -85*n^2 -374*n -960)/32: n in [0..30]]; // _G. C. Greubel_, May 15 2019
%o A053528 (Sage) [binomial(n,5)*(n^5 +5*n^4 +5*n^3 -85*n^2 -374*n -960)/32 for n in (0..30)] # _G. C. Greubel_, May 15 2019
%o A053528 (GAP) List([0..30], n-> Binomial(n,5)*(n^5 +5*n^4 +5*n^3 -85*n^2 -374*n -960)/32) # _G. C. Greubel_, May 15 2019
%Y A053528 Column k=5 of A117279.
%Y A053528 Cf. A000217 (1 edge), A050534 (2 edges), A053526 (3 edges), A053527 (4 edges).
%K A053528 nonn,easy
%O A053528 0,6
%A A053528 _N. J. A. Sloane_, Jan 16 2000