OFFSET
0,2
COMMENTS
Triangle T(n,k), 0 <= k <= n, read by rows given by [3, -2, 0, 0, 0, 0, 0, ...] DELTA [3, -2, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 07 2007
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..5150 (rows n = 0..100, flattened)
FORMULA
3*Pascal's triangle A007318, then replace T(0,0) with 1.
G.f.: Sum_{n>=0} Sum_{k>=0} T(n,k) *x^n * y^k = 1 - 3*(1+y)*x/(x+x*y-1). - R. J. Mathar, Feb 19 2020
From G. C. Greubel, Apr 27 2021: (Start)
T(n, k) = 3*binomial(n,k) - 2*[n=0].
Sum_{k=0..n} T(n, k) = 3*2^n - 2*[n=0] = A082505(n+1). (End)
E.g.f.: 3*exp(x*(1+y)) - 2. - Stefano Spezia, Apr 03 2024
EXAMPLE
First few rows of the triangle:
1;
3, 3;
3, 6, 3;
3, 9, 9, 3;
3, 12, 18, 12, 3;
3, 15, 30, 30, 15, 3;
3, 18, 45, 60, 45, 18, 3;
...
MATHEMATICA
Join[{1}, Rest[Flatten[Table[3Binomial[n, k], {n, 0, 10}, {k, 0, n}]]]] (* Harvey P. Dale, Feb 15 2014 *)
Table[3*Binomial[n, k] -2*Boole[n==0], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 26 2021 *)
PROG
(Magma)
A134059:= func< n, k | n eq 0 select 1 else 3*Binomial(n, k) >;
[A134059(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 26 2021
(Sage)
def A134059(n, k): return 3*binomial(n, k) - 2*bool(n==0)
flatten([[A134059(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 26 2021
CROSSREFS
KEYWORD
AUTHOR
Gary W. Adamson, Oct 05 2007
EXTENSIONS
Title changed by G. C. Greubel, Apr 26 2021
STATUS
approved