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A355262 Array of Fuss-Catalan numbers read by ascending antidiagonals, A(n, k) = binomial(k*n + 1, k)/(k*n + 1).

%I A355262 #26 Sep 29 2024 09:19:47
%S A355262 1,1,1,1,1,0,1,1,1,0,1,1,2,1,0,1,1,3,5,1,0,1,1,4,12,14,1,0,1,1,5,22,
%T A355262 55,42,1,0,1,1,6,35,140,273,132,1,0,1,1,7,51,285,969,1428,429,1,0,1,1,
%U A355262 8,70,506,2530,7084,7752,1430,1,0
%N A355262 Array of Fuss-Catalan numbers read by ascending antidiagonals, A(n, k) = binomial(k*n + 1, k)/(k*n + 1).
%C A355262 An alternative definition is: the Fuss-Catalan sequences (A(n, k), k >= 0 ) are the main diagonals of the Fuss-Catalan triangles of order n - 1. See A355173 for the definition of a Fuss-Catalan triangle.
%D A355262 N. I. Fuss, Solutio quaestionis, quot modis polygonum n laterum in polygona m laterum, per diagonales resolvi queat, Nova Acta Academiae Scientiarum Imperialis Petropolitanae, vol.9 (1791), 243-251.
%D A355262 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 1990, (Eqs. 5.70, 7.66, and sec. 7.5, example 5).
%H A355262 Per Alexandersson, Frether Getachew Kebede, Samuel Asefa Fufa, and Dun Qiu, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Getachew/get3.html">Pattern-Avoidance and Fuss-Catalan Numbers</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.4.2.
%H A355262 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, pp. 337-338.
%H A355262 Jean-Luc Baril, Mireille Bousquet-Mélou, Sergey Kirgizov, and Mehdi Naima, <a href="https://arxiv.org/abs/2409.15982">The ascent lattice on Dyck paths</a>, arXiv:2409.15982 [math.CO], 2024. See p. 6.
%H A355262 Jean-Christophe Aval, <a href="https://arxiv.org/abs/0711.0906">Multivariate Fuss-Catalan Numbers</a>, arXiv:0711.0906v1, Discrete Math., 308 (2008), 4660-4669.
%H A355262 Thomas A. Dowling, <a href="https://docplayer.net/6827054-Catalan-numbers-thomas-a-dowling-department-of-mathematics-ohio-state-uni-versity.html">Catalan Numbers</a>, Chapter 7.
%H A355262 D. E. Knuth, <a href="https://arxiv.org/abs/math/9207221">Convolution polynomials</a>, arXiv:math/9207221 [math.CA]; Mathematica J. 2.1 (1992), no. 4, 67-78.
%H A355262 Donald Knuth's 20th Annual Christmas Tree Lecture, <a href="https://www.youtube.com/watch?v=P4AaGQIo0HY">(3/2)-ary Trees</a>, Stanford Online, Video 2014.
%H A355262 Wojciech Młotkowski, <a href="https://www.math.uni-bielefeld.de/documenta/vol-15/28.pdf">Fuss-Catalan Numbers in Noncommutative Probability</a>, Docum. Math. 15:939-955, (2010).
%H A355262 Wikipedia, <a href="https://en.wikipedia.org/wiki/Fuss%E2%80%93Catalan_number">Fuss-Catalan number</a>.
%F A355262 A(n, k) = (1/k!) * Product_{j=1..k-1} (k*n + 1 - j).
%F A355262 A(n, k) = (binomial(k*n, k) + binomial(k*n, k-1)) / (k*n + 1).
%F A355262 Let B(t, z) = Sum_{k>=0} binomial(k*t + 1, k)*z^k / (k*t + 1), then
%F A355262 A(n, k) = [z^k] B(n, z).
%e A355262 Array A(n, k) begins:
%e A355262 [0] 1, 1, 0,   0,    0,     0,      0,       0,         0, ...  A019590
%e A355262 [1] 1, 1, 1,   1,    1,     1,      1,       1,         1, ...  A000012
%e A355262 [2] 1, 1, 2,   5,   14,    42,    132,     429,      1430, ...  A000108
%e A355262 [3] 1, 1, 3,  12,   55,   273,   1428,    7752,     43263, ...  A001764
%e A355262 [4] 1, 1, 4,  22,  140,   969,   7084,   53820,    420732, ...  A002293
%e A355262 [5] 1, 1, 5,  35,  285,  2530,  23751,  231880,   2330445, ...  A002294
%e A355262 [6] 1, 1, 6,  51,  506,  5481,  62832,  749398,   9203634, ...  A002295
%e A355262 [7] 1, 1, 7,  70,  819, 10472, 141778, 1997688,  28989675, ...  A002296
%e A355262 [8] 1, 1, 8,  92, 1240, 18278, 285384, 4638348,  77652024, ...  A007556
%e A355262 [9] 1, 1, 9, 117, 1785, 29799, 527085, 9706503, 184138713, ...  A062994
%p A355262 A := (n, k) -> binomial(k*n + 1, k)/(k*n + 1):
%p A355262 for n from 0 to 9 do seq(A(n, k), k = 0..8) od;
%t A355262 (* See the Knuth references. In the christmas lecture Knuth has fun calculating the Fuss-Catalan development of Pi and i. *)
%t A355262 B[t_, n_] := Sum[Binomial[t k+1, k] z^k / (t k+1), {k, 0, n-1}] + O[z]^n
%t A355262 Table[CoefficientList[B[n, 9], z], {n, 0, 9}] // TableForm
%Y A355262 Cf. A091144 (main diagonal), A019590, A000012, A000108, A001764, A002293, A002294, A002295, A002296, A007556, A062994.
%Y A355262 Variants: A062993, A070914.
%Y A355262 Fuss-Catalan triangles: A123110 (order 0), A355173 (order 1), A355172 (order 2), A355174 (order 3).
%K A355262 nonn,tabl
%O A355262 0,13
%A A355262 _Peter Luschny_, Jun 26 2022