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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

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

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

Cf. A028326, A134059, A173048, A173049.

Sequence in context: A292929 A049627 A278223 * A345530 A216955 A086973

Adjacent sequences:  A134055 A134056 A134057 * A134059 A134060 A134061

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 September 27 03:05 EDT 2021. Contains 347673 sequences. (Running on oeis4.)