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%I #15 Jan 09 2020 18:14:33
%S 1,1,1,1,2,2,1,3,6,5,1,4,10,20,14,1,5,14,35,70,42,1,6,18,50,126,252,
%T 132,1,7,22,65,182,462,924,429,1,8,26,80,238,672,1716,3432,1430,1,9,
%U 30,95,294,882,2508,6435,12870,4862,1,10,34,110,350,1092,3300,9438,24310,48620,16796
%N Array read by descending antidiagonals: A(n,k) = (1 + k*n)*C(n) where C(n) = Catalan numbers (A000108).
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%H Andrew Howroyd, <a href="/A330965/b330965.txt">Table of n, a(n) for n = 0..1325</a>
%F A(n,k) = (1 + k*n)*binomial(2*n,n)/(n+1).
%F A(n,k) = 2*(k*n+1)*(2*n-1)*A(n-1,k)/((n+1)*(k*n-k+1)) for n > 0.
%F G.f. of column k: (k - 1 - (2*k-4)*x - (k-1)*sqrt(1 - 4*x))/(2*x*sqrt(1 - 4*x)).
%e Array begins:
%e ====================================================
%e n\k | 0 1 2 3 4 5 6 7
%e ----+-----------------------------------------------
%e 0 | 1 1 1 1 1 1 1 1 ...
%e 1 | 1 2 3 4 5 6 7 8 ...
%e 2 | 2 6 10 14 18 22 26 30 ...
%e 3 | 5 20 35 50 65 80 95 110 ...
%e 4 | 14 70 126 182 238 294 350 406 ...
%e 5 | 42 252 462 672 882 1092 1302 1512 ...
%e 6 | 132 924 1716 2508 3300 4092 4884 5676 ...
%e 7 | 429 3432 6435 9438 12441 15444 18447 21450 ...
%e ...
%o (PARI) T(n, k)={(1 + k*n)*binomial(2*n,n)/(n+1)}
%Y Columns k=0..12 are A000108, A000984, A001700, A051924, A051944, A051945, A050476, A050477, A050478, A050479, A050489, A050490, A050491.
%K nonn,tabl
%O 0,5
%A _Andrew Howroyd_, Jan 04 2020
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