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A237619 Riordan array (1/(1+x*c(x)), x*c(x)) where c(x) is the g.f. of Catalan numbers (A000108).

%I A237619 #10 May 28 2022 04:02:48
%S A237619 1,-1,1,0,0,1,-1,1,1,1,-2,2,3,2,1,-6,6,8,6,3,1,-18,18,24,18,10,4,1,
%T A237619 -57,57,75,57,33,15,5,1,-186,186,243,186,111,54,21,6,1,-622,622,808,
%U A237619 622,379,193,82,28,7,1,-2120,2120,2742,2120,1312,690,311,118,36,8,1
%N A237619 Riordan array (1/(1+x*c(x)), x*c(x)) where c(x) is the g.f. of Catalan numbers (A000108).
%H A237619 G. C. Greubel, <a href="/A237619/b237619.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A237619 Sum_{k=0..n} T(n,k)*x^k = A126983(n), A000957(n+1), A026641(n) for x = 0, 1, 2 respectively.
%F A237619 T(n, k) = A167772(n-1, k-1) for k > 0, with T(n, 0) = A167772(n, 0).
%F A237619 T(n, 0) = A126983(n).
%F A237619 T(n+1, 1) = A000957(n+1).
%F A237619 T(n+2, 2) = A000958(n+1).
%F A237619 T(n+3, 3) = A104629(n) = A000957(n+3).
%F A237619 T(n+4, 4) = A001558(n).
%F A237619 T(n+5, 5) = A001559(n).
%F A237619 T(n, k) = A065602(n, k) for k > 0, with T(n, k) = (-1)^(n-k), for n < 2, and T(n, 0) = A065602(n, 0). - _G. C. Greubel_, May 27 2022
%e A237619 Triangle begins:
%e A237619     1;
%e A237619    -1,  1;
%e A237619     0,  0,  1;
%e A237619    -1,  1,  1,  1;
%e A237619    -2,  2,  3,  2,  1;
%e A237619    -6,  6,  8,  6,  3,  1;
%e A237619   -18, 18, 24, 18, 10,  4, 1;
%e A237619   -57, 57, 75, 57, 33, 15, 5, 1;
%e A237619 Production matrix begins:
%e A237619   -1, 1;
%e A237619   -1, 1, 1;
%e A237619   -1, 1, 1, 1;
%e A237619   -1, 1, 1, 1, 1;
%e A237619   -1, 1, 1, 1, 1, 1;
%e A237619   -1, 1, 1, 1, 1, 1, 1;
%e A237619   -1, 1, 1, 1, 1, 1, 1, 1;
%e A237619   -1, 1, 1, 1, 1, 1, 1, 1, 1;
%t A237619 A065602[n_, k_]:= A065602[n, k]= Sum[(k-1+2*j)*Binomial[2*(n-j)-k-1, n-1]/(2*(n - j) -k-1), {j,0,(n-k)/2}];
%t A237619 T[n_, k_]:= If[k==0, A065602[n, 0], If[n==1 && k==1, 1, A065602[n, k]]];
%t A237619 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, May 27 2022 *)
%o A237619 (SageMath)
%o A237619 def A065602(n, k): return sum( (k+2*j-1)*binomial(2*n-2*j-k-1, n-1)/(2*n-2*j-k-1) for j in (0..(n-k)//2) )
%o A237619 def A237619(n, k):
%o A237619     if (n<2): return (-1)^(n-k)
%o A237619     elif (k==0): return A065602(n, 0)
%o A237619     else: return A065602(n, k)
%o A237619 flatten([[A237619(n, k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 27 2022
%Y A237619 Diagonals: A000012, A000217, A023443, A166830.
%Y A237619 Cf. A000957, A000958, A001558, A001559, A104629, A126983.
%Y A237619 Cf. A065602, A167772.
%K A237619 sign,tabl
%O A237619 0,11
%A A237619 _Philippe Deléham_, Feb 10 2014