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A155839 A ratio of two Catalan arrays.

%I A155839 #8 Sep 08 2022 08:45:41
%S A155839 1,0,1,0,0,1,0,1,0,1,0,2,3,0,1,0,4,7,6,0,1,0,8,18,16,10,0,1,0,16,45,
%T A155839 51,30,15,0,1,0,32,110,152,115,50,21,0,1,0,64,264,436,396,225,77,28,0,
%U A155839 1,0,128,624,1212,1300,876,399,112,36,0,1
%N A155839 A ratio of two Catalan arrays.
%H A155839 G. C. Greubel, <a href="/A155839/b155839.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A155839 T(n, k) = Sum_{j=k..n} (-1)^(n-j)*binomial(j+1, n-j)*binomial(j, k)*A000108(j-k).
%F A155839 Sum_{k=0..n} T(n, k) = A120010(n+1).
%F A155839 Equals A033184^{-1}*A124644.
%e A155839 Triangle begins
%e A155839   1;
%e A155839   0,  1;
%e A155839   0,  0,   1;
%e A155839   0,  1,   0,   1;
%e A155839   0,  2,   3,   0,   1;
%e A155839   0,  4,   7,   6,   0,  1;
%e A155839   0,  8,  18,  16,  10,  0,  1;
%e A155839   0, 16,  45,  51,  30, 15,  0, 1;
%e A155839   0, 32, 110, 152, 115, 50, 21, 0, 1;
%t A155839 T[n_, k_] = Sum[(-1)^j*Binomial[n-j, k]*Binomial[n-j+1, j]*CatalanNumber[n-k-j], {j, 0, n-k}];
%t A155839 Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Jun 04 2021 *)
%o A155839 (Magma)
%o A155839 A155839:= func< n,k | (&+[(-1)^(n-j)*Binomial(j+1, n-j)*Binomial(j, k)*Catalan(j-k) : j in [k..n]]) >;
%o A155839 [A155839(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 04 2021
%o A155839 (Sage)
%o A155839 def A155839(n,k): return sum( (-1)^j*binomial(n-j,k)*binomial(n-j+1,j)*catalan_number(n-k-j) for j in (0..n-k))
%o A155839 flatten([[A155839(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 04 2021
%Y A155839 Cf. A000108, A033184, A120010 (row sums), A124644.
%K A155839 easy,nonn,tabl
%O A155839 0,12
%A A155839 _Paul Barry_, Jan 28 2009