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Triangle read by rows, (n-1) zeros followed by (2n, 1).
9

%I #16 Mar 25 2022 02:29:08

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

%C A134082 * [1,2,3,...] = A084849: (1, 4, 11, 22, 37, ...).

%C Binomial transform of A134082 = A134083.

%C A112295 replaces subdiagonal with (-1,-3,-5, ...).

%H G. C. Greubel, <a href="/A134082/b134082.txt">Rows n = 0..50 of the triangle, flattened</a>

%F 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 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 Row polynomial e.g.f.: exp(x*y)(1+2x). - _Tom Copeland_, Dec 03 2013

%F Sum_{k=0..n} T(n,k) = 2*n+1 = A005408(n). - _G. C. Greubel_, Feb 17 2021

%e First few rows of the triangle:

%e 1;

%e 2, 1;

%e 0, 4, 1;

%e 0, 0, 6, 1;

%e 0, 0, 0, 8, 1;

%e 0, 0, 0, 0, 10, 1;

%e ...

%t T[n_, k_]:= If[k==n, 1, If[k==n-1, 2*n, 0]];

%t Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 17 2021 *)

%o (Sage)

%o def A134082(n,k): return 1 if k==n else 2*n if k==n-1 else 0

%o flatten([[A134082(n,k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Feb 17 2021

%o (Magma)

%o A134082:= func< n,k | k eq n select 1 else k eq n-1 select 2*n else 0 >;

%o [A134082(n,k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Feb 17 2021

%Y Cf. A005408, A084849, A112295, A134083.

%K nonn,tabl

%O 0,2

%A _Gary W. Adamson_, Oct 07 2007

%E More terms added by _G. C. Greubel_, Feb 17 2021