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A330140 Triangle read by rows: binomial transform of a signed variant of triangle A026300 with alternating signs in each column.

%I A330140 #19 Sep 08 2022 08:46:24
%S A330140 1,0,1,0,0,2,0,0,1,4,0,0,1,4,9,0,0,1,5,15,21,0,0,1,6,24,50,51,0,0,1,7,
%T A330140 35,98,161,127,0,0,1,8,48,168,378,504,323,0,0,1,9,63,264,750,1386,
%U A330140 1554,835,0,0,1,10,80,390,1335,3132,4920,4740,2188
%N A330140 Triangle read by rows: binomial transform of a signed variant of triangle A026300 with alternating signs in each column.
%e A330140 The signed variant of triangle A026300 begins:
%e A330140    1;
%e A330140   -1,   1;
%e A330140    1,  -2,   2;
%e A330140   -1,   3,  -5,   4;
%e A330140    1,  -4,   9, -12,   9;
%e A330140   ...
%e A330140 The binomial transform of the foregoing is as shown below.
%e A330140 Written as a triangle the sequence begins:
%e A330140   1;
%e A330140   0,  1;
%e A330140   0,  0,  2;
%e A330140   0,  0,  1,  4;
%e A330140   0,  0,  1,  4,  9;
%e A330140   0,  0,  1,  5, 15, 21;
%e A330140   0,  0,  1,  6, 24, 50, 51;
%e A330140   ...
%t A330140 F[n_, k_]:= (-1)^(n+k)*Sum[Binomial[n, 2i+n-k]*(Binomial[2i+n-k, i] - Binomial[2i+n-k, i-1]), {i,0,Floor[k/2]}]; T[n_, k_]:= Sum[Binomial[n, j]*F[j, k], {j,k,n}]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jan 06 2020 *)
%o A330140 (Magma)
%o A330140 F:= func< n,k | (-1)^(n+k)*&+[Binomial(n,2*i+n-k)*(Binomial(2*i+n-k,i) - Binomial(2*i+n-k,i-1)): i in [0..Floor(k/2)]] >;
%o A330140 T:= func< n,k | &+[Binomial(n,j)*F(j,k): j in [k..n]] >;
%o A330140 [T(n,k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Jan 06 2020
%Y A330140 Right border gives A001006, the Motzkin numbers.
%Y A330140 Row sums give A000108, the Catalan numbers.
%Y A330140 Cf. A026300, A007318, A330141, A329959.
%K A330140 nonn,tabl
%O A330140 0,6
%A A330140 _Gary W. Adamson_, Dec 02 2019