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A134058 Triangle read by rows, T(n,k) = 2*binomial(n,k) if k>0, (0<=k<=n), left column = (1,2,2,2,...). 7
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

Row sums = (1, 4, 8, 16, 32, 64,...). A134059 = analogous triangle, replacing (1,2,2,2,...) with (1,3,3,3,...).

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

Table of n, a(n) for n=0..54.

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<=k<=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

EXAMPLE

First few rows of the triangle are:

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, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 09 2015, after Vladimir Kruchinin *)

CROSSREFS

Cf. A134059.

Sequence in context: A255336 A049627 A278223 * A216955 A086973 A240131

Adjacent sequences:  A134055 A134056 A134057 * A134059 A134060 A134061

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Oct 05 2007

STATUS

approved

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Last modified September 24 21:29 EDT 2017. Contains 292441 sequences.