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A134082 Triangle read by rows, (n-1) zeros followed by (2n, 1).

%I A134082 #16 Mar 25 2022 02:29:08
%S A134082 1,2,1,0,4,1,0,0,6,1,0,0,0,8,1,0,0,0,0,10,1,0,0,0,0,0,12,1,0,0,0,0,0,
%T A134082 0,14,1,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,20,
%U A134082 1,0,0,0,0,0,0,0,0,0,0,22,1,0,0,0,0,0,0,0,0,0,0,0,24,1,0,0,0,0,0,0,0,0,0,0,0,0,26,1
%N A134082 Triangle read by rows, (n-1) zeros followed by (2n, 1).
%C A134082 A134082 * [1,2,3,...] = A084849: (1, 4, 11, 22, 37, ...).
%C A134082 Binomial transform of A134082 = A134083.
%C A134082 A112295 replaces subdiagonal with (-1,-3,-5, ...).
%H A134082 G. C. Greubel, <a href="/A134082/b134082.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A134082 Triangle read by rows, (n-1) zeros followed by (2n, 1). As an infinite lower triangular matrix, (1,1,1,...) in the main diagonal and (2,4,6,8,...) in the subdiagonal.
%F A134082 From formalism in A132382, e.g.f. = I_o[2*(u*x)^(1/2)] (1+2x) where I_o is the zeroth modified Bessel function of the first kind, i.e., I_o[2*(u*x)^(1/2)] = Sum_{j>=0} u^j/j! * x^j/j!. - _Tom Copeland_, Dec 07 2007
%F A134082 Row polynomial e.g.f.: exp(x*y)(1+2x). - _Tom Copeland_, Dec 03 2013
%F A134082 Sum_{k=0..n} T(n,k) = 2*n+1 = A005408(n). - _G. C. Greubel_, Feb 17 2021
%e A134082 First few rows of the triangle:
%e A134082   1;
%e A134082   2,  1;
%e A134082   0,  4,  1;
%e A134082   0,  0,  6,  1;
%e A134082   0,  0,  0,  8,  1;
%e A134082   0,  0,  0,  0, 10,  1;
%e A134082   ...
%t A134082 T[n_, k_]:= If[k==n, 1, If[k==n-1, 2*n, 0]];
%t A134082 Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 17 2021 *)
%o A134082 (Sage)
%o A134082 def A134082(n,k): return 1 if k==n else 2*n if k==n-1 else 0
%o A134082 flatten([[A134082(n,k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Feb 17 2021
%o A134082 (Magma)
%o A134082 A134082:= func< n,k | k eq n select 1 else k eq n-1 select 2*n else 0 >;
%o A134082 [A134082(n,k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Feb 17 2021
%Y A134082 Cf. A005408, A084849, A112295, A134083.
%K A134082 nonn,tabl
%O A134082 0,2
%A A134082 _Gary W. Adamson_, Oct 07 2007
%E A134082 More terms added by _G. C. Greubel_, Feb 17 2021