cp's OEIS Frontend

A062994 Eighth column of triangle A062993 (without leading zeros). A Pfaff-Fuss or 9-Raney sequence.

%I A062994 #46 Jun 16 2025 08:47:16
%S A062994 1,1,9,117,1785,29799,527085,9706503,184138713,3573805950,70625252863,
%T A062994 1416298046436,28748759731965,589546754316126,12195537924351375,
%U A062994 254184908607118800,5332692942907262361
%N A062994 Eighth column of triangle A062993 (without leading zeros). A Pfaff-Fuss or 9-Raney sequence.
%C A062994 See Graham et al., Hilton and Pedersen, Hoggat and Bicknell, Frey and Sellers references given in A062993.
%C A062994 Essentially the same as A059967. a(n), n>=1, enumerates 9-ary trees (rooted, ordered, incomplete) with n vertices (including the root).
%C A062994 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 A062994 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 A062994 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 A062994 Harry J. Smith, <a href="/A062994/b062994.txt">Table of n, a(n) for n = 0..100</a>
%H A062994 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 A062994 a(n) = A062993(n+7, 7) = binomial(9*n, n)/(8*n+1).
%F A062994 G.f.: RootOf((_Z^9)*x-_Z+1) (Maple notation, from ECS, see links for A007556).
%F A062994 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 A062994 From _Ilya Gutkovskiy_, Jan 16 2017: (Start)
%F A062994 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 A062994 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 A062994 a(n) ~ 3^(18*n+1)/(sqrt(Pi)*2^(24*n+5)*n^(3/2)). (End)
%F A062994 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
%F A062994 G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^17). - _Seiichi Manyama_, Jun 16 2025
%e A062994 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 A062994 seq(binomial(9*k+1,k)/(8*k+1),k=0..30);
%p A062994 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 A062994 Table[Binomial[9n,n]/(8n+1),{n,0,30}] (* _Harvey P. Dale_, Oct 28 2012 *)
%o A062994 (PARI) { for (n=0, 100, write("b062994.txt", n, " ", binomial(9*n, n)/(8*n + 1)) ) } \\ _Harry J. Smith_, Aug 15 2009
%Y A062994 Column k=8 of A070914.
%Y A062994 Cf. A000108, A001764, A002293, A002294, A002295, A002296, A007556, A059968, A130564.
%K A062994 nonn,easy
%O A062994 0,3
%A A062994 _Wolfdieter Lang_, Jul 12 2001
%E A062994 9-ary tree comments and Pólya and G. Szegő reference from _Wolfdieter Lang_, Sep 14 2007