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Eighth column of triangle A062993 (without leading zeros). A Pfaff-Fuss or 9-Raney sequence.
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%I #43 Feb 05 2024 08:24:02

%S 1,1,9,117,1785,29799,527085,9706503,184138713,3573805950,70625252863,

%T 1416298046436,28748759731965,589546754316126,12195537924351375,

%U 254184908607118800,5332692942907262361

%N Eighth column of triangle A062993 (without leading zeros). A Pfaff-Fuss or 9-Raney sequence.

%C See Graham et al., Hilton and Pedersen, Hoggat and Bicknell, Frey and Sellers references given in A062993.

%C Essentially the same as A059967. a(n), n>=1, enumerates 9-ary trees (rooted, ordered, incomplete) with n vertices (including the root).

%C These numbers appear in a formula on p. 24 of Gross et al. for b = -2 or 4. For b = -1 or 3, see A002293.- _Tom Copeland_, Dec 24 2019

%C This is instance k = 9 of the generalized Catalan family {C(k, n)}_{n>=0} given in a comment of A130564. - Wolfdieter Lang, Feb 05 2024

%D G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, Heidelberg, New York, 2 vols., 1972, Vol. 1, problem. 211, p. 146 with solution on p. 348.

%H Harry J. Smith, <a href="/A062994/b062994.txt">Table of n, a(n) for n = 0..100</a>

%H M. Gross, P. Hacking, S. Keel, and M. Kontsevich, <a href="https://arxiv.org/abs/1411.1394">Canonical bases for cluster algebras</a>, arXiv preprint arXiv:1411.1394 [math.AG], 2016.

%F a(n) = A062993(n+7, 7) = binomial(9*n, n)/(8*n+1).

%F G.f.: RootOf((_Z^9)*x-_Z+1) (Maple notation, from ECS, see links for A007556).

%F Recurrence: a(0) = 1; a(n) = Sum_{i1+i2+..+i9=n-1} a(i1)*a(i2)*...*a(i9) for n>=1. - _Robert FERREOL_, Apr 01 2015

%F From _Ilya Gutkovskiy_, Jan 16 2017: (Start)

%F O.g.f.: 8F7(1/9,2/9,1/3,4/9,5/9,2/3,7/9,8/9; 1/4,3/8,1/2,5/8,3/4,7/8,9/8; 387420489*x/16777216).

%F E.g.f.: 8F8(1/9,2/9,1/3,4/9,5/9,2/3,7/9,8/9; 1/4,3/8,1/2,5/8,3/4,7/8,1,9/8; 387420489*x/16777216).

%F a(n) ~ 3^(18*n+1)/(sqrt(Pi)*2^(24*n+5)*n^(3/2)). (End)

%F D-finite with recurrence: 128*n*(8*n-5)*(4*n-1)*(8*n+1)*(2*n-1)*(8*n-1)*(4*n-3)*(8*n-3)*a(n) -81*(9*n-7)*(9*n-5)*(3*n-1)*(9*n-1)*(9*n-8)*(3*n-2)*(9*n-4)*(9*n-2)*a(n-1)=0. - _R. J. Mathar_, Feb 20 2020

%e There are a(2)=9 9-ary trees (vertex degree <=9 and 9 possible branchings) with 2 vertices (one of them the root). Adding one more branch (one more vertex) to these 9 trees yields 9*9 + binomial(9,2) = 117 = a(3) such trees.

%p seq(binomial(9*k+1,k)/(8*k+1),k=0..30);

%p n:=30: G:=series(RootOf(g = 1+x*g^9, g),x=0,n+1): seq(coeff(G,x,k),k=0..n); # _Robert FERREOL_, Apr 01 2015

%t Table[Binomial[9n,n]/(8n+1),{n,0,30}] (* _Harvey P. Dale_, Oct 28 2012 *)

%o (PARI) { for (n=0, 100, write("b062994.txt", n, " ", binomial(9*n, n)/(8*n + 1)) ) } \\ _Harry J. Smith_, Aug 15 2009

%Y Column k=8 of A070914.

%Y Cf. A000108, A001764, A002293, A002294, A002295, A002296, A007556, A059968, A130564.

%K nonn,easy

%O 0,3

%A _Wolfdieter Lang_, Jul 12 2001

%E 9-ary tree comments and Pólya and G. Szegő reference from _Wolfdieter Lang_, Sep 14 2007