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A295679 Array read by antidiagonals: T(n,k) = k-Modular Catalan numbers C_{n,k} (n >= 0, k > 0).

%I A295679 #35 Oct 14 2022 05:35:39
%S A295679 1,1,1,1,1,1,1,1,2,1,1,1,2,4,1,1,1,2,5,8,1,1,1,2,5,13,16,1,1,1,2,5,14,
%T A295679 35,32,1,1,1,2,5,14,41,96,64,1,1,1,2,5,14,42,124,267,128,1,1,1,2,5,14,
%U A295679 42,131,384,750,256,1,1,1,2,5,14,42,132,420,1210,2123,512,1
%N A295679 Array read by antidiagonals: T(n,k) = k-Modular Catalan numbers C_{n,k} (n >= 0, k > 0).
%C A295679 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 A295679 Theorem: C_{n,k} enumerates the following objects:
%C A295679 (1) binary trees with n internal nodes avoiding a certain subtree (i.e., comb_k^{+1}),
%C A295679 (2) plane trees with n+1 nodes whose non-root nodes have degree less than k,
%C A295679 (3) Dyck paths of length 2n avoiding a down-step followed immediately by k consecutive up-steps,
%C A295679 (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 A295679 (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 A295679 (6) permutations of {1,2,...,n} avoiding 1-3-2 and 23...(k+1)1.
%C A295679 Columns of the array converge rowwise to A000108. The diagonal k=n-1 is A001453. - _Andrey Zabolotskiy_, Dec 02 2017
%H A295679 Andrew Howroyd, <a href="/A295679/b295679.txt">Table of n, a(n) for n = 0..1274</a>
%H A295679 Nickolas Hein, Jia Huang, <a href="https://doi.org/10.1016/j.ejc.2016.11.004">Modular Catalan Numbers</a>, European Journal of Combinatorics, 61 (2017), 197-218, <a href="http://arxiv.org/abs/1508.01688">arXiv:1508.01688</a> [math.CO], 2015-2016.
%F A295679 G.f. of column k: 1/(1-G(x)) where G(x) is the reversion of x*(1-x)/(1-x^k).
%e A295679 Array begins (n >= 0, k > 0):
%e A295679 ======================================================
%e A295679 n\k| 1   2    3    4    5    6    7    8    9   10
%e A295679 ---|--------------------------------------------------
%e A295679 0  | 1   1    1    1    1    1    1    1    1    1 ...
%e A295679 1  | 1   1    1    1    1    1    1    1    1    1 ...
%e A295679 2  | 1   2    2    2    2    2    2    2    2    2 ...
%e A295679 3  | 1   4    5    5    5    5    5    5    5    5 ...
%e A295679 4  | 1   8   13   14   14   14   14   14   14   14 ...
%e A295679 5  | 1  16   35   41   42   42   42   42   42   42 ...
%e A295679 6  | 1  32   96  124  131  132  132  132  132  132 ...
%e A295679 7  | 1  64  267  384  420  428  429  429  429  429 ...
%e A295679 8  | 1 128  750 1210 1375 1420 1429 1430 1430 1430 ...
%e A295679 9  | 1 256 2123 3865 4576 4796 4851 4861 4862 4862 ...
%e A295679 ...
%p A295679 A295679 := proc(n,k)
%p A295679     if n = 0 then
%p A295679         1;
%p A295679     else
%p A295679         add((-1)^j/n*binomial(n,j)*binomial(2*n-j*k,n+1),j=0..(n-1)/k) ;
%p A295679     end if ;
%p A295679 end proc:
%p A295679 seq(seq( A295679(n,d-n),n=0..d-1),d=1..12) ; # _R. J. Mathar_, Oct 14 2022
%t A295679 rows = cols = 12;
%t A295679 col[k_] := Module[{G}, G = InverseSeries[x*(1-x)/(1-x^k) + O[x]^cols, x]; CoefficientList[1/(1 - G), x]];
%t A295679 A = Array[col, cols];
%t A295679 T[n_, k_] := A[[k, n+1]];
%t A295679 Table[T[n-k+1, k], {n, 0, rows-1}, {k, n+1, 1, -1}] // Flatten (* _Jean-François Alcover_, Dec 05 2017, adapted from PARI *)
%o A295679 (PARI)
%o A295679 T(n,k)=polcoeff(1/(1-serreverse(x*(1-x)/(1-x^k) + O(x^max(2,n+1)))), n);
%o A295679 for(n=0, 10, for(k=1, 10, print1(T(n, k), ", ")); print);
%Y A295679 Columns 3..9 are A005773, A159772, A261588, A261589, A261590, A261591, A261592.
%Y A295679 Cf. A288942, A000108, A001453.
%K A295679 nonn,tabl,easy
%O A295679 0,9
%A A295679 _Andrew Howroyd_, Nov 30 2017