A134059 - OEIS

%I #43 Apr 04 2024 10:14:38

%S 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,

%T 3,21,63,105,105,63,21,3,3,24,84,168,210,168,84,24,3,3,27,108,252,378,

%U 378,252,108,27,3

%N Triangle T(n, k) = 3*binomial(n,k) with T(0, 0) = 1, read by rows.

%C 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

%H Vincenzo Librandi, <a href="/A134059/b134059.txt">Table of n, a(n) for n = 0..5150</a> (rows n = 0..100, flattened)

%F 3*Pascal's triangle A007318, then replace T(0,0) with 1.

%F 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

%F From _G. C. Greubel_, Apr 27 2021: (Start)

%F T(n, k) = 3*binomial(n,k) - 2*[n=0].

%F Sum_{k=0..n} T(n, k) = 3*2^n - 2*[n=0] = A082505(n+1). (End)

%F E.g.f.: 3*exp(x*(1+y)) - 2. - _Stefano Spezia_, Apr 03 2024

%e First few rows of the triangle:

%e   1;

%e   3,  3;

%e   3,  6,  3;

%e   3,  9,  9,  3;

%e   3, 12, 18, 12,  3;

%e   3, 15, 30, 30, 15,  3;

%e   3, 18, 45, 60, 45, 18,  3;

%e   ...

%t Join[{1},Rest[Flatten[Table[3Binomial[n,k],{n,0,10},{k,0,n}]]]] (* _Harvey P. Dale_, Feb 15 2014 *)

%t Table[3*Binomial[n,k] -2*Boole[n==0], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Apr 26 2021 *)

%o (Magma)

%o A134059:= func< n,k | n eq 0 select 1 else 3*Binomial(n,k) >;

%o [A134059(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Apr 26 2021

%o (Sage)

%o def A134059(n,k): return 3*binomial(n,k) - 2*bool(n==0)

%o flatten([[A134059(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Apr 26 2021

%Y Cf. A007318, A082505 (row sums), A084938, A134058.

%K nonn,tabl,easy

%O 0,2

%A _Gary W. Adamson_, Oct 05 2007

%E Title changed by _G. C. Greubel_, Apr 26 2021