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A329959
Binomial transform of a signed variant of triangle A050166.
2
1, 0, 2, 0, 0, 5, 0, 0, 1, 14, 0, 0, 1, 8, 42, 0, 0, 1, 10, 45, 132, 0, 0, 1, 12, 69, 220, 429, 0, 0, 1, 14, 98, 406, 1001, 1430, 0, 0, 1, 16, 132, 672, 2184, 4368, 4862, 0, 0, 1, 18, 171, 1032, 4152, 11088, 18564, 16796, 0, 0, 1, 20, 215, 1500, 7185, 23904, 54060, 77520, 58786
OFFSET
0,3
COMMENTS
Row sums = A007317(n+1).
Right border = A000108(n+1).
LINKS
FORMULA
T(n,k) = Sum_{j=k..n} C(n,j) * (-1)^(j+k) * A050166(j,k). - Alois P. Heinz, Nov 27 2019
EXAMPLE
The signed variant of A050166 is A050166(n,k) * (-1)^(n+k):
1;
-1, 2;
1, -4, 5;
-1, 6, -14, 14;
1, -8, 27, -48, 42;
...
Let the above triangle = S, and Pascal's triangle = P as an infinite lower triangular matrix. Then T = P * S gives:
1;
0, 2;
0, 0, 5;
0, 0, 1, 14;
0, 0, 1, 8, 42;
0, 0, 1, 10, 45, 132;
...
MAPLE
S:= (n, k)-> (binomial(2*n, k)-binomial(2*n, k-2))*(-1)^(n+k):
T:= (n, k)-> add(binomial(n, j)*S(j, k), j=k..n):
seq(seq(T(n, k), k=0..n), n=0..10); # Alois P. Heinz, Nov 27 2019
MATHEMATICA
Table[Sum[(-1)^(k+j)*Binomial[n, j]*(Binomial[2*j, k] - Binomial[2*j, k-2]), {j, k, n}], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Jan 06 2020 *)
PROG
(PARI) T(n, k) = sum(j=k, n, (-1)^(k+j)*binomial(n, j)*(binomial(2*j, k) - binomial(2*j, k-2)) ); \\ G. C. Greubel, Jan 06 2020
(Magma) T:= func< n, k | &+[(-1)^(k+j)*Binomial(n, j)*(Binomial(2*j, k) - Binomial(2*j, k-2)): j in [k..n]] >;
[T(n, k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jan 06 2020
(Sage) [[sum((-1)^(k+j)*binomial(n, j)*(binomial(2*j, k) - binomial(2*j, k-2)) for j in (k..n)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jan 06 2020
(GAP) B:=Binomial;; Flat(List([0..10], n-> List([0..n], k-> Sum([k..n], j-> (-1)^(k+j)*B(n, j)*(B(2*j, k) - B(2*j, k-2)) )))); # G. C. Greubel, Jan 06 2020
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Nov 25 2019
EXTENSIONS
New offset and more terms from Alois P. Heinz, Nov 25 2019
STATUS
approved