A069996 Number of spanning trees on the bipartite graph K_{3,n}.

0, 1, 12, 81, 432, 2025, 8748, 35721, 139968, 531441, 1968300, 7144929, 25509168, 89813529, 312487308, 1076168025, 3673320192, 12440502369, 41841412812, 139858796529, 464904586800, 1537671920841, 5062810950252, 16600580533161, 54226471004352, 176518460300625

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

This is the second binomial transform of the octagonal numbers A000567 and the binomial transform of A084857. - Paul Barry, Jun 09 2003

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Wrath of Math, 2025 is a Strange Number, YouTube video, 2025.
• Index entries for linear recurrences with constant coefficients, signature (9,-27,27).

Crossrefs

Cf. A000567, A084857.

Programs

Mathematica

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

Formula

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

E.g.f.: exp(3*x)*(x+3*x^2). - Paul Barry, Jul 23 2003

From Colin Barker, Aug 10 2012: (Start)

a(n) = 9*a(n-1)-27*a(n-2)+27*a(n-3).

G.f.: x*(1+3*x)/(1-3*x)^3. (End)

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

Extensions

Edited and extended by Robert G. Wilson v, May 04 2002

a(0)=0 prepended by Alois P. Heinz, Jan 03 2025