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%I #61 Mar 12 2024 12:18:01
%S 1,3,12,45,162,567,1944,6561,21870,72171,236196,767637,2480058,
%T 7971615,25509168,81310473,258280326,817887699,2582803260,8135830269,
%U 25569752274,80196041223,251048476872,784526490225,2447722649502
%N Number of ternary trees (A001764) with n nodes and maximal diameter.
%C A problem important for polymer science because it counts the trees having unbranched branches; they are called "combs".
%C Equals (1, 3, 9, 27, 81, ...) convolved with (1, 0, 3, 9, 27, 81, ...). Example: a(5) = 162 = (81, 27, 9, 3, 1) dot (1, 0, 3, 9, 27) = 81 + 3*27. - _Gary W. Adamson_, Jul 31 2010
%C Floretion Algebra Multiplication Program, FAMP Code: lesforseq[ - 'i + 'j - 'kk' - 'ki' - 'kj' ], vesforseq(n) = 3^n, tesforseq = A006234
%H Harry J. Smith, <a href="/A064017/b064017.txt">Table of n, a(n) for n = 1..200</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-9).
%F a(n) = 3*a(n-1) + 3^(n-2).
%F a(n) = (n+1)*3^(n-2), for n > 1.
%F From _Paul Barry_, Sep 05 2003: (Start)
%F a(n) = (n+2)3^(n-1) + 0^n/3 (offset 0).
%F a(n) = A025192(n) + A027471(n). (End)
%F A006234(n+4) - a(n+2) = 3^n. - _Creighton Dement_, Mar 01 2005
%F a(n+1) = Sum_{k=0..n} A196389(n,k)*3^k. - _Philippe Deléham_, Oct 31 2011
%F G.f.: (1 - 3*x + 3*x^2)*x/(1 - 3*x)^2. - _Philippe Deléham_, Oct 31 2011
%F a(n) = 6*a(n-1) - 9*a(n-2), with a(1)=1, a(2)=3, a(3)=12. - _Harvey P. Dale_, Feb 07 2012
%F E.g.f.: (exp(3*x)*(1 + 3*x) - 1)/9. - _Stefano Spezia_, Mar 05 2020
%F From _Amiram Eldar_, Jan 18 2021: (Start)
%F Sum_{n>=1} 1/a(n) = 27*log(3/2) - 19/2.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 17/2 - 27*log(4/3). (End)
%e a(5) = 162 because we can write (5+1)*3^(5-2) = 6*3^3 = 6*27.
%p a:=n->ceil(sum(3^(n-2),j=0..n)): seq(a(n), n=1..26); # _Zerinvary Lajos_, Jun 05 2008
%t Join[{1},Table[(n+1)3^(n-2),{n,2,30}]] (* or *) Join[{1}, LinearRecurrence[ {6,-9},{3,12},30]] (* _Harvey P. Dale_, Feb 07 2012 *)
%o (PARI) { for (n=1, 200, if (n>1, a=(n + 1)*p; p*=3, a=p=1); write("b064017.txt", n, " ", a) ) } \\ _Harry J. Smith_, Sep 06 2009
%o (PARI) a(n)=if(n==1, 1, (n+1)*3^(n-2)); \\ _Joerg Arndt_, May 06 2013
%o (SageMath)
%o @CachedFunction
%o def BB(n, k, x): # modified cardinal B-splines
%o if n == 1: return 0 if (x < 0) or (x >= k) else 1
%o return x*BB(n-1, k, x) + (n*k-x)*BB(n-1, k, x-k)
%o def EulerianPolynomial(n, k, x):
%o if n == 0: return 1
%o return add(BB(n+1, k, k*m+1)*x^m for m in (0..n))
%o def A064017(n) : return 3^(n-1)*EulerianPolynomial(1,n-1,1/3) if n != 1 else 1
%o [A064017(n) for n in (1..25)] # _Peter Luschny_, May 04 2013
%Y Cf. A001764, A006234, A014915, A025192, A027261, A079272, A196389.
%K nonn,nice,easy
%O 1,2
%A Danail Bonchev (bonchevd(AT)aol.com), Sep 07 2001
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