This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A134058 #26 Apr 04 2024 10:14:29
%S A134058 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 A134058 14,42,70,70,42,14,2,2,16,56,112,140,112,56,16,2,2,18,72,168,252,252,
%U A134058 168,72,18,2
%N A134058 Triangle T(n, k) = 2*binomial(n, k) with T(0, 0) = 1, read by rows.
%C A134058 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 A134058 Equals A028326 for all but the first term. - _R. J. Mathar_, Jun 08 2008
%C A134058 Warning: the row sums do not give A046055. - _N. J. A. Sloane_, Jul 08 2009
%H A134058 G. C. Greubel, <a href="/A134058/b134058.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A134058 Double Pascal's triangle and replace leftmost column with (1,2,2,2,...).
%F A134058 M*A007318, where M = an infinite lower triangular matrix with (1,2,2,2,...) in the main diagonal and the rest zeros.
%F A134058 Sum_{k=0..n} T(n,k) = A151821(n+1). - _Philippe Deléham_, Sep 17 2009
%F A134058 G.f.: (1+x+y)/(1-x-y). - _Vladimir Kruchinin_, Apr 09 2015
%F A134058 T(n, k) = 2*binomial(n, k) - [n=0]. - _G. C. Greubel_, Apr 26 2021
%F A134058 E.g.f.: 2*exp(x*(1+y)) - 1. - _Stefano Spezia_, Apr 03 2024
%e A134058 First few rows of the triangle:
%e A134058 1
%e A134058 2, 2;
%e A134058 2, 4, 2;
%e A134058 2, 6, 6, 2;
%e A134058 2, 8, 12, 8, 2;
%e A134058 2, 10, 20, 20, 10, 2;
%e A134058 ...
%t A134058 T[n_, k_]:= SeriesCoefficient[(1+x+y)/(1-x-y), {x, 0, n-k}, {y, 0, k}];
%t A134058 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _Jean-François Alcover_, Apr 09 2015, after _Vladimir Kruchinin_ *)
%t A134058 Table[2*Binomial[n,k] -Boole[n==0], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Apr 26 2021 *)
%o A134058 (Magma)
%o A134058 A134058:= func< n,k | n eq 0 select 1 else 2*Binomial(n,k) >;
%o A134058 [A134058(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Apr 26 2021
%o A134058 (Sage)
%o A134058 def A134058(n,k): return 2*binomial(n,k) - bool(n==0)
%o A134058 flatten([[A134058(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Apr 26 2021
%Y A134058 Cf. A007318, A028326, A046055, A084938, A134059, A151821, A173048, A173049.
%K A134058 nonn,tabl
%O A134058 0,2
%A A134058 _Gary W. Adamson_, Oct 05 2007
%E A134058 Title changed by _G. C. Greubel_, Apr 26 2021