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%I #8 Aug 29 2017 04:52:24
%S 1,2,5,14,43,142,497,1828,7037,28326,119361,527748,2454929,12041410,
%T 62354641,340840118,1963757863,11896370734,75549183725,501393978466,
%U 3467199478543,24916100775758,185646100106929,1431332539961350
%N Row sums of a triangle based on the Catalan numbers.
%C Row sums of A110488.
%H G. C. Greubel, <a href="/A110489/b110489.txt">Table of n, a(n) for n = 0..595</a>
%F a(n) = Sum_{k=0..n} Sum_{j=0..(n-k)} 2*(j+1)*(k-1)^j*C(2*(n-k)+1, n-k-j)/ (n-k+j+2).
%t T[n_, 0] := CatalanNumber[n]; T[n_, 1] := CatalanNumber[n]; T[n_, n_] := 1; T[n_, k_] := Sum[2*(j + 1)*(k - 1)^j*Binomial[2 (n - k) + 1, n - k - j]/(n - k + j + 2), {j, 0, n - k}]; Table[Sum[T[n, k], {k, 0, n}], {n, 0, 50}] (* _G. C. Greubel_, Aug 29 2017 *)
%o (PARI) a(n) = sum(k=0, n, sum(j=0, (n-k), 2*(j+1)*(k-1)^j*binomial(2*(n-k)+1, n-k-j)/ (n-k+j+2))); \\ _Michel Marcus_, Aug 29 2017
%Y Cf. A110488.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Jul 22 2005