OFFSET
0,2
COMMENTS
Triangle T(n,k), 0 <= k <= n, read by rows, given by [2, -1, 0, 0, 0, 0, 0, ...] DELTA [2, -1, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 07 2007
Equals A028326 for all but the first term. - R. J. Mathar, Jun 08 2008
Warning: the row sums do not give A046055. - N. J. A. Sloane, Jul 08 2009
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
Double Pascal's triangle and replace leftmost column with (1,2,2,2,...).
M*A007318, where M = an infinite lower triangular matrix with (1,2,2,2,...) in the main diagonal and the rest zeros.
Sum_{k=0..n} T(n,k) = A151821(n+1). - Philippe Deléham, Sep 17 2009
G.f.: (1+x+y)/(1-x-y). - Vladimir Kruchinin, Apr 09 2015
T(n, k) = 2*binomial(n, k) - [n=0]. - G. C. Greubel, Apr 26 2021
E.g.f.: 2*exp(x*(1+y)) - 1. - Stefano Spezia, Apr 03 2024
EXAMPLE
First few rows of the triangle:
1
2, 2;
2, 4, 2;
2, 6, 6, 2;
2, 8, 12, 8, 2;
2, 10, 20, 20, 10, 2;
...
MATHEMATICA
T[n_, k_]:= SeriesCoefficient[(1+x+y)/(1-x-y), {x, 0, n-k}, {y, 0, k}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* Jean-François Alcover, Apr 09 2015, after Vladimir Kruchinin *)
Table[2*Binomial[n, k] -Boole[n==0], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 26 2021 *)
PROG
(Magma)
A134058:= func< n, k | n eq 0 select 1 else 2*Binomial(n, k) >;
[A134058(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 26 2021
(Sage)
def A134058(n, k): return 2*binomial(n, k) - bool(n==0)
flatten([[A134058(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 26 2021
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Oct 05 2007
EXTENSIONS
Title changed by G. C. Greubel, Apr 26 2021
STATUS
approved