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A330140
Triangle read by rows: binomial transform of a signed variant of triangle A026300 with alternating signs in each column.
1
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, 35, 98, 161, 127, 0, 0, 1, 8, 48, 168, 378, 504, 323, 0, 0, 1, 9, 63, 264, 750, 1386, 1554, 835, 0, 0, 1, 10, 80, 390, 1335, 3132, 4920, 4740, 2188
OFFSET
0,6
EXAMPLE
The signed variant of triangle A026300 begins:
1;
-1, 1;
1, -2, 2;
-1, 3, -5, 4;
1, -4, 9, -12, 9;
...
The binomial transform of the foregoing is as shown below.
Written as a triangle the sequence begins:
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;
...
MATHEMATICA
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 *)
PROG
(Magma)
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)]] >;
T:= func< n, k | &+[Binomial(n, j)*F(j, k): j in [k..n]] >;
[T(n, k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jan 06 2020
CROSSREFS
Right border gives A001006, the Motzkin numbers.
Row sums give A000108, the Catalan numbers.
Sequence in context: A342243 A073429 A123634 * A091866 A168511 A111146
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Dec 02 2019
STATUS
approved