OFFSET
1,1
COMMENTS
With T(n,0) = n for n >= 0, this becomes J. M. Bergot's triangular array in the definition of A190730. - Petros Hadjicostas, Feb 15 2021
LINKS
Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of the triangle, flattened)
FORMULA
T(n, k) = (n + 1)*2^k - 1 for n >= 1 and 1 <= k <= n.
From Petros Hadjicostas, Feb 15 2021: (Start)
Sum_{k=1..n} T(n,k) = A190730(n).
T(n,2) = 4*n + 3 for n >= 2.
T(n,n) = A087323(n).
T(n,n-1) = A099035(n) = (n+1)*2^(n-1) - 1 for n >= 2.
Recurrence: T(n,k) = 3*T(n,k-1) - 2*T(n,k-2) for n >= 2 and 2 <= k <= n with initial conditions the values of T(n, 1) and T(n,2).
Bivariate o.g.f.: Sum_{n,k>=1} T(n,k)*x^n*y^k = (4*x^3*y^2 - 2*x^2*y - 4*x*y - x + 3)*x*y/((1 - 2*x*y)^2*(1 - x*y)*(1 - x)^2). (End)
EXAMPLE
Triangle T(n,k) (with rows n >= 1 and columns k = 1..n) begins:
3;
5, 11;
7, 15, 31;
9, 19, 39, 79;
11, 23, 47, 95, 191;
13, 27, 55, 111, 223, 447;
15, 31, 63, 127, 255, 511, 1023;
17, 35, 71, 143, 287, 575, 1151, 2303;
19, 39, 79, 159, 319, 639, 1279, 2559, 5119;
...
MATHEMATICA
A087322row[n_]:=NestList[2#+1&, 2n+1, n-1]; Array[A087322row, 10] (* Paolo Xausa, Oct 17 2023 *)
CROSSREFS
KEYWORD
AUTHOR
Amarnath Murthy, Sep 03 2003
EXTENSIONS
Edited and extended by David Wasserman, May 06 2005
Name edited by Petros Hadjicostas, Feb 15 2021
STATUS
approved