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A076038 Square array read by ascending antidiagonals in which row n has g.f. C/(1-n*x*C) where C = (1/2-1/2*(1-4*x)^(1/2))/x = g.f. for Catalan numbers A000108.

%I A076038 #16 Sep 04 2025 01:36:44
%S A076038 1,1,1,1,2,2,1,3,5,5,1,4,10,14,14,1,5,17,35,42,42,1,6,26,74,126,132,
%T A076038 132,1,7,37,137,326,462,429,429,1,8,50,230,726,1446,1716,1430,1430,1,
%U A076038 9,65,359,1434,3858,6441,6435,4862,4862,1,10,82,530,2582,8952,20532,28770,24310,16796,16796
%N A076038 Square array read by ascending antidiagonals in which row n has g.f. C/(1-n*x*C) where C = (1/2-1/2*(1-4*x)^(1/2))/x = g.f. for Catalan numbers A000108.
%F A076038 A(n, m) = 1/(m+1)*Sum_{k=0..m} binomial(2*m-k, m)*(k+1)*(n-m)^k, m=0..n.
%e A076038 Array begins as:
%e A076038   1 1  2  5  14  42 ... (n=0)
%e A076038   1 2  5 14  42 132 ... (n=1)
%e A076038   1 3 10 35 126 ... (n=2)
%e A076038   1 4 17 74 326 ...
%e A076038   ...
%t A076038 Unprotect[Power]; Power[0,0]=1; Protect[Power]; A[n_, m_]:= 1/(m+1)*Sum[Binomial[2*m-k, m]*(k+1)*(n-m)^k,{k,0,m}]; Table[A[n,m],{n,0,10},{m,0,n}]//Flatten (* _Stefano Spezia_, Sep 01 2025 *)
%Y A076038 Rows give A000108, A000108, A001700, A049027, A076025, A076026. Cf. A076037, A067347.
%Y A076038 Cf. A059718.
%K A076038 nonn,tabl,changed
%O A076038 0,5
%A A076038 _N. J. A. Sloane_, Oct 29 2002
%E A076038 More terms from _Vladeta Jovovic_, Jul 18 2003
%E A076038 a(63)-a(65) from _Stefano Spezia_, Sep 01 2025