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Number of spanning trees on the bipartite graph K_{3,n}.
7

%I #17 Mar 23 2024 07:25:24

%S 1,12,81,432,2025,8748,35721,139968,531441,1968300,7144929,25509168,

%T 89813529,312487308,1076168025,3673320192,12440502369,41841412812,

%U 139858796529,464904586800,1537671920841,5062810950252,16600580533161

%N Number of spanning trees on the bipartite graph K_{3,n}.

%C With a leading zero, this is the second binomial transform of the octagonal numbers A000567 and the binomial transform of A084857. - _Paul Barry_, Jun 09 2003

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (9,-27,27).

%F a(n) = n^2 * 3^(n-1).

%F E.g.f.: exp(3x)(x+3x^2). - _Paul Barry_, Jul 23 2003

%F a(n) = 9*a(n-1)-27*a(n-2)+27*a(n-3). G.f.: x*(1+3*x)/(1-3*x)^3. - _Colin Barker_, Aug 10 2012

%F G.f.: 1 + 12*x/(G(0) - 12*x), where G(k)= 1 + 12*x + 2*k*(6*x+1) + (1+3*x)*k^2 - 3*x*(k+1)^2*(k+3)^2/G(k+1) ; (continued fraction). - _Sergei N. Gladkovskii_, Jul 05 2013

%t a[n_] := n^2*3^(n - 1); Table[ a[n], {n, 1, 24}]

%Y Cf. A000567, A084857.

%K nonn,easy

%O 1,2

%A Eric Weinhandl (eweinhandl(AT)msn.com), May 01 2002

%E Edited and extended by _Robert G. Wilson v_, May 04 2002