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A134058 Triangle T(n, k) = 2*binomial(n, k) with T(0, 0) = 1, read by rows. 10
1, 2, 2, 2, 4, 2, 2, 6, 6, 2, 2, 8, 12, 8, 2, 2, 10, 20, 20, 10, 2, 2, 12, 30, 40, 30, 12, 2, 2, 14, 42, 70, 70, 42, 14, 2, 2, 16, 56, 112, 140, 112, 56, 16, 2, 2, 18, 72, 168, 252, 252, 168, 72, 18, 2 (list; table; graph; refs; listen; history; text; internal format)
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
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
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
Sequence in context: A049627 A278223 A368556 * A345530 A216955 A086973
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Oct 05 2007
EXTENSIONS
Title changed by G. C. Greubel, Apr 26 2021
STATUS
approved

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Last modified March 28 10:55 EDT 2024. Contains 371241 sequences. (Running on oeis4.)