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A328902 Triangle T(n, k) read by rows: T(n, k) is the denominator of the rational Catalan number defined as binomial(n + k, n)/(n + k) for 0 <= k <= n, n > 0; T(0, 0) = 1.

%I A328902 #27 Oct 31 2020 03:46:43
%S A328902 1,1,1,2,1,2,3,1,1,3,4,1,2,1,4,5,1,1,1,1,5,6,1,2,3,1,1,1,7,1,1,1,1,1,
%T A328902 1,7,8,1,2,1,4,1,2,1,8,9,1,1,3,1,1,3,1,1,9,10,1,2,1,2,5,2,1,1,1,5,11,
%U A328902 1,1,1,1,1,1,1,1,1,1,11,12,1,2,3,4,1,3,1,2,3,1,1,6
%N A328902 Triangle T(n, k) read by rows: T(n, k) is the denominator of the rational Catalan number defined as binomial(n + k, n)/(n + k) for 0 <= k <= n, n > 0; T(0, 0) = 1.
%H A328902 Stefano Spezia, <a href="/A328902/b328902.txt">First 141 rows of the triangle, flattened</a>
%H A328902 D. Armstrong, N. A. Loehr, G. S. Warrington, <a href="https://doi.org/10.1007/s00026-015-0293-6">Rational Parking Functions and Catalan Numbers</a>, Annals of Combinatorics (2016), Volume 20, Issue 1, pp 21-58.
%H A328902 M. T. L. Bizley, <a href="http://bergeron.math.uqam.ca/wp-content/uploads/2014/09/Bizley.pdf">Derivation of a new formula for the number of minimal lattice paths from (0, 0) to (km, kn) having just t contacts with the line my = nx and having no points above this line; and a proof of Grossman's formula for the number of paths which may touch but do not rise above this line</a>, Journal of the Institute of Actuaries, Vol. 80, No. 1 (1954): 55-62.
%F A328902 T(n, k) = (n + k)/gcd(binomial(n + k, n), n + k) for n > 0.
%e A328902 n\k| 0 1 2 3 4 5 6
%e A328902 ---+--------------
%e A328902 0  | 1
%e A328902 1  | 1 1
%e A328902 2  | 2 1 2
%e A328902 3  | 3 1 1 3
%e A328902 4  | 4 1 2 1 4
%e A328902 5  | 5 1 1 1 1 5
%e A328902 6  | 6 1 2 3 1 1 1
%e A328902 ...
%t A328902 Flatten[Join[{1},Table[(n+k)/GCD[n+k,Binomial[n+k,n]],{n,1,12},{k,0,n}]]]
%o A328902 (PARI) A328902(n,k)=if(n,(n+k)/gcd(binomial(n+k,n),n+k),1) \\ _M. F. Hasler_, Nov 04 2019
%Y A328902 Cf. A000108, A028310 (1st column), A046899, A051162, A328901 (numerator).
%K A328902 frac,nonn,tabl
%O A328902 0,4
%A A328902 _Stefano Spezia_, Oct 30 2019