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Triangle T(n, k) = 2*binomial(n, k) with T(0, 0) = 1, read by rows.
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%I #26 Apr 04 2024 10:14:29

%S 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,

%T 14,42,70,70,42,14,2,2,16,56,112,140,112,56,16,2,2,18,72,168,252,252,

%U 168,72,18,2

%N Triangle T(n, k) = 2*binomial(n, k) with T(0, 0) = 1, read by rows.

%C 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

%C Equals A028326 for all but the first term. - _R. J. Mathar_, Jun 08 2008

%C Warning: the row sums do not give A046055. - _N. J. A. Sloane_, Jul 08 2009

%H G. C. Greubel, <a href="/A134058/b134058.txt">Rows n = 0..50 of the triangle, flattened</a>

%F Double Pascal's triangle and replace leftmost column with (1,2,2,2,...).

%F M*A007318, where M = an infinite lower triangular matrix with (1,2,2,2,...) in the main diagonal and the rest zeros.

%F Sum_{k=0..n} T(n,k) = A151821(n+1). - _Philippe Deléham_, Sep 17 2009

%F G.f.: (1+x+y)/(1-x-y). - _Vladimir Kruchinin_, Apr 09 2015

%F T(n, k) = 2*binomial(n, k) - [n=0]. - _G. C. Greubel_, Apr 26 2021

%F E.g.f.: 2*exp(x*(1+y)) - 1. - _Stefano Spezia_, Apr 03 2024

%e First few rows of the triangle:

%e 1

%e 2, 2;

%e 2, 4, 2;

%e 2, 6, 6, 2;

%e 2, 8, 12, 8, 2;

%e 2, 10, 20, 20, 10, 2;

%e ...

%t T[n_, k_]:= SeriesCoefficient[(1+x+y)/(1-x-y), {x, 0, n-k}, {y, 0, k}];

%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _Jean-François Alcover_, Apr 09 2015, after _Vladimir Kruchinin_ *)

%t Table[2*Binomial[n,k] -Boole[n==0], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Apr 26 2021 *)

%o (Magma)

%o A134058:= func< n,k | n eq 0 select 1 else 2*Binomial(n,k) >;

%o [A134058(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Apr 26 2021

%o (Sage)

%o def A134058(n,k): return 2*binomial(n,k) - bool(n==0)

%o flatten([[A134058(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Apr 26 2021

%Y Cf. A007318, A028326, A046055, A084938, A134059, A151821, A173048, A173049.

%K nonn,tabl

%O 0,2

%A _Gary W. Adamson_, Oct 05 2007

%E Title changed by _G. C. Greubel_, Apr 26 2021