OFFSET
0,4
COMMENTS
LINKS
G. C. Greubel, Rows n=0..100 of triangle, flattened
FORMULA
Number triangle T(n, k) = Sum_{i=0..n} (-1)^(n-i)*binomial(n+i-k, i-k).
Riordan array (1/(1-x*c(x)-2*x^2*c(x)^2), x*c(x)) where c(x)=g.f. of A000108.
The production matrix M (discarding the zeros) is:
1, 1;
3, 1, 1;
3, 1, 1, 1;
3, 1, 1, 1, 1;
... such that the n-th row of the triangle is the top row of M^n. - Gary W. Adamson, Feb 16 2012
EXAMPLE
Rows begin
1;
1,1;
4,2,1;
13,7,3,1;
46,24,11,4,1;
166,86,40,16,5,1;
MATHEMATICA
Table[Sum[(-1)^(n-j)*Binomial[n+j-k, j-k], {j, 0, n}], {n, 0, 12}, {k, 0, n}] //Flatten (* G. C. Greubel, Feb 19 2019 *)
PROG
(PARI) {T(n, k) = sum(j=0, n, (-1)^(n-j)*binomial(n+j-k, j-k))};
for(n=0, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Feb 19 2019
(Magma) [[(&+[(-1)^(n-j)*Binomial(n+j-k, j-k): j in [0..n]]): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Feb 19 2019
(Sage) [[sum((-1)^(n-j)*binomial(n+j-k, j-k) for j in (0..n)) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Feb 19 2019
(GAP) Flat(List([0..12], n-> List([0..n], k-> Sum([0..n], j-> (-1)^(n-j)*Binomial(n+j-k, j-k) )))); # G. C. Greubel, Feb 19 2019
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Jun 23 2005
STATUS
approved