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

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, 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, 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

Offset

0,2

Comments

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

Binomial transform of A134082 = A134083.

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

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

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.

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

Row polynomial e.g.f.: exp(x*y)(1+2x). - Tom Copeland, Dec 03 2013

Sum_{k=0..n} T(n,k) = 2*n+1 = A005408(n). - G. C. Greubel, Feb 17 2021

Example

First few rows of the triangle:
  1;
  2,  1;
  0,  4,  1;
  0,  0,  6,  1;
  0,  0,  0,  8,  1;
  0,  0,  0,  0, 10,  1;
  ...

Mathematica

T[n_, k_]:= If[k==n, 1, If[k==n-1, 2*n, 0]];
Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 17 2021 *)

Prog

(Sage)
def A134082(n, k): return 1 if k==n else 2*n if k==n-1 else 0
flatten([[A134082(n, k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Feb 17 2021
(Magma)
A134082:= func< n, k | k eq n select 1 else k eq n-1 select 2*n else 0 >;
[A134082(n, k): k in [0..n], n in [0..15]]; // G. C. Greubel, Feb 17 2021

Crossrefs

Cf. A005408, A084849, A112295, A134083.

Sequence in context: A053117 A121448 A019094 * A185740 A139360 A326759

Adjacent sequences: A134079 A134080 A134081 * A134083 A134084 A134085

Keyword

nonn,tabl

Author

Gary W. Adamson, Oct 07 2007

Extensions

More terms added by G. C. Greubel, Feb 17 2021

Status

approved