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 A233832 a(n) = 2*binomial(7*n+2,n)/(7*n+2). 7
 1, 2, 15, 154, 1827, 23562, 320866, 4540200, 66096459, 983592304, 14894775896, 228784720710, 3555866673450, 55819631671902, 883738853546472, 14094715154157680, 226245021605612955, 3652242142988400570, 59254515909624764575, 965678197027521177200 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=7, r=2. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906v1, Discrete Math., 308 (2008), 4660-4669. Thomas A. Dowling, Catalan Numbers Chapter 7 Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k, arXiv:2204.14023 [math.CO], 2022. Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955. Wikipedia, Fuss-Catalan number Sheng-liang Yang and Mei-yang Jiang, Pattern avoiding problems on the hybrid d-trees, J. Lanzhou Univ. Tech., (China, 2023) Vol. 49, No. 2, 144-150. (in Mandarin) FORMULA G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=7, r=2. a(n) = 2*binomial(7n+1, n-1)/n for n>0, a(0)=1. [Bruno Berselli, Jan 19 2014] From Ilya Gutkovskiy, Sep 14 2018: (Start) E.g.f.: 6F6(2/7,3/7,4/7,5/7,6/7,8/7; 1/2,2/3,5/6,1,7/6,4/3; 823543*x/46656). a(n) ~ 7^(7*n+3/2)/(sqrt(Pi)*3^(6*n+5/2)*4^(3*n+1)*n^(3/2)). (End) MATHEMATICA Table[2 Binomial[7 n + 2, n]/(7 n + 2), {n, 0, 30}] PROG (PARI) a(n) = 2*binomial(7*n+2, n)/(7*n+2); (PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(7/2))^2+x*O(x^n)); polcoeff(B, n)} (Magma) [2*Binomial(7*n+2, n)/(7*n+2): n in [0..30]]; CROSSREFS Cf. A000108, A002296, A233833 - A233835, A143547, A130565, A233907, A233908. Sequence in context: A191364 A308379 A373357 * A185756 A362364 A239107 Adjacent sequences: A233829 A233830 A233831 * A233833 A233834 A233835 KEYWORD nonn AUTHOR Tim Fulford, Dec 16 2013 STATUS approved

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