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A261591 8-Modular Catalan Numbers C_{n,8}.

%I A261591 #25 Mar 07 2020 11:43:25
%S A261591 1,1,2,5,14,42,132,429,1430,4861,16784,58695,207452,739840,2658936,
%T A261591 9620232,35011566,128082523,470731970,1737220254,6435115168,
%U A261591 23918062480,89172805980,333396591075,1249717509612,4695654554206,17682176062376,66720743308877
%N A261591 8-Modular Catalan Numbers C_{n,8}.
%C A261591 Definition: Given a primitive k-th root of unity w, a binary operation a*b=a+wb, and sufficiently general fixed complex numbers x_0, ..., x_n, the k-modular Catalan numbers C_{n,k} enumerate parenthesizations of x_0*x_1*...*x_n that give distinct values.
%C A261591 Theorem: C_{n,k} enumerates the following objects:
%C A261591 (1) binary trees with n internal nodes avoiding a certain subtree (i.e., comb_k^{+1}),
%C A261591 (2) plane trees with n+1 nodes whose non-root nodes have degree less than k,
%C A261591 (3) Dyck paths of length 2n avoiding a down-step followed immediately by k consecutive up-steps,
%C A261591 (4) partitions with n nonnegative parts bounded by the staircase partition (n-1,n-2,...,1,0) such that each positive number appears fewer than k times,
%C A261591 (5) standard 2-by-n Young tableaux whose top row avoids contiguous labels of the form i,j+1,j+2,...,j+k for all i<j, and
%C A261591 (6) permutations of {1,2,...,n} avoiding 1-3-2 and 23...(k+1)1.
%H A261591 Andrew Howroyd, <a href="/A261591/b261591.txt">Table of n, a(n) for n = 0..200</a>
%H A261591 Nickolas Hein, Jia Huang, <a href="http://arxiv.org/abs/1508.01688">Modular Catalan Numbers</a>, arXiv:1508.01688 [math.CO], 2016-2016.
%F A261591 sum( 1<=l<=n, (l/n)sum( m_1+...+m_k=n and m_2+2m_3+...+(k-1)m_k=n-l , MC(n;m_1,...,m_k) ) ), where MC(n;m_1,...,m_k) is the multinomial coefficient associated to the multiset (m_1,...,m_k).
%F A261591 G.f.: 1/(1-x*G(x)) where G(x) is g.f. of A036769. - _Andrew Howroyd_, Dec 04 2017
%e A261591 The Catalan number C_9=4862 counts the parenthesizations of x_1*...*x_10 where * is arbitrary. Defining * and w as above and writing x_i compactly as xi, we have x1*(x2*(x3*(x4*(x5*(x6*(x7*(x8*(x9*(x10))))))))) = x1+wx2+w^2x3+w^3x4+w^4x5+w^5x6+w^6x7+w^7x8+x9+wx10 = x1*(x2*(x3*(x4*(x5*(x6*(x7*(x8*(x9))))))))*(x10). For n=9 and k=8, these are the only parenthesizations that give the same value for x1*...*x10, so C_{9,8}=4862-1=4861.
%t A261591 terms = 30; col[k_] := Module[{G}, G = InverseSeries[x*(1 - x)/(1 - x^k) + O[x]^terms, x]; CoefficientList[1/(1 - G), x]];
%t A261591 col[8] (* _Jean-François Alcover_, Dec 05 2017, after _Andrew Howroyd_ *)
%o A261591 (PARI) Vec(1/(1-serreverse(x*(1-x)/(1-x^8) + O(x*x^25)))) \\ _Andrew Howroyd_, Dec 04 2017
%o A261591 (Sage)
%o A261591 def C(k):
%o A261591     print(1)
%o A261591     for n in range(1,51):
%o A261591         f = ((1-x^k)/(1-x))^n # ((x+1)^2-x^2*(x/(x+1))^(k-2))^n
%o A261591         f = f.simplify_full()
%o A261591         C = 0
%o A261591         for i in range(n):
%o A261591             C = C + (n-i)*f.coefficient(x,i)/n
%o A261591         print(C)
%o A261591 time C(8)
%Y A261591 Column k=8 of A295679.
%Y A261591 C_{n,1} is the all 1's sequence A000012. For C_{n,k} with k=2,3,4 see A011782, A005773, A159772. For k=5,6,7,9 see A261588, A261589, A261590, A261592.
%Y A261591 Cf. A036769.
%K A261591 nonn
%O A261591 0,3
%A A261591 _Nickolas Hein_, Aug 25 2015