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A330965 Array read by descending antidiagonals: A(n,k) = (1 + k*n)*C(n) where C(n) = Catalan numbers (A000108).

%I A330965 #20 Aug 24 2025 11:30:12
%S A330965 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 A330965 132,1,7,22,65,182,462,924,429,1,8,26,80,238,672,1716,3432,1430,1,9,
%U A330965 30,95,294,882,2508,6435,12870,4862,1,10,34,110,350,1092,3300,9438,24310,48620,16796
%N A330965 Array read by descending antidiagonals: A(n,k) = (1 + k*n)*C(n) where C(n) = Catalan numbers (A000108).
%D A330965 A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%H A330965 Andrew Howroyd, <a href="/A330965/b330965.txt">Table of n, a(n) for n = 0..1325</a>
%F A330965 A(n,k) = (1 + k*n)*binomial(2*n,n)/(n+1).
%F A330965 A(n,k) = 2*(k*n+1)*(2*n-1)*A(n-1,k)/((n+1)*(k*n-k+1)) for n > 0.
%F A330965 G.f. of column k: (k - 1 - (2*k-4)*x - (k-1)*sqrt(1 - 4*x))/(2*x*sqrt(1 - 4*x)).
%e A330965 Array begins:
%e A330965 ====================================================
%e A330965 n\k |   0    1    2    3     4     5     6     7
%e A330965 ----+-----------------------------------------------
%e A330965   0 |   1    1    1    1     1     1     1     1 ...
%e A330965   1 |   1    2    3    4     5     6     7     8 ...
%e A330965   2 |   2    6   10   14    18    22    26    30 ...
%e A330965   3 |   5   20   35   50    65    80    95   110 ...
%e A330965   4 |  14   70  126  182   238   294   350   406 ...
%e A330965   5 |  42  252  462  672   882  1092  1302  1512 ...
%e A330965   6 | 132  924 1716 2508  3300  4092  4884  5676 ...
%e A330965   7 | 429 3432 6435 9438 12441 15444 18447 21450 ...
%e A330965   ...
%t A330965 A330965[n_, k_] := CatalanNumber[n]*(k*n + 1);
%t A330965 Table[A330965[k, n - k], {n, 0, 10}, {k, 0, n}] (* _Paolo Xausa_, Aug 24 2025 *)
%o A330965 (PARI) T(n, k)={(1 + k*n)*binomial(2*n,n)/(n+1)}
%Y A330965 Columns k=0..12 are A000108, A000984, A001700, A051924, A051944, A051945, A050476, A050477, A050478, A050479, A050489, A050490, A050491.
%K A330965 nonn,tabl,changed
%O A330965 0,5
%A A330965 _Andrew Howroyd_, Jan 04 2020