login
Numbers in (4,4)-Pascal triangle .
3

%I #7 May 03 2021 20:13:56

%S 1,4,4,4,8,4,4,12,12,4,4,16,24,16,4,4,20,40,40,20,4,4,24,60,80,60,24,

%T 4,4,28,84,140,140,84,28,4,4,32,112,224,280,224,112,32,4,4,36,144,336,

%U 504,504,336,144,36,4,4,40,180,480,840,1008,840,480,180,40,4

%N Numbers in (4,4)-Pascal triangle .

%C This triangle belongs to the family of (x,y)-Pascal triangles ; other triangles arise by choosing different values for (x,y): (1,1) -> A007318 ; (1,0) -> A071919 ; (3,2) -> A029618 ; (2,2) -> A134058 ; (-1,1) -> A112467 ; (0,1) -> A097805 ; (5,5) -> A135089 ; etc..

%H G. C. Greubel, <a href="/A132200/b132200.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n,k) = 4*binomial(n,k), n>0 ; T(0,0)=1.

%F Sum_{k=0..n} T(n,k) = 2^(n+2) - 3*[n=0]. - _G. C. Greubel_, May 03 2021

%e Triangle begins:

%e 1;

%e 4, 4;

%e 4, 8, 4;

%e 4, 12, 12, 4;

%e 4, 16, 24, 16, 4;

%e 4, 20, 40, 40, 20, 4;

%t Table[4*Binomial[n,k] -3*Boole[n==0], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, May 03 2021 *)

%o (Magma) [1] cat [4*Binomial(n,k): k in [0..n], n in [1..12]]; // _G. C. Greubel_, May 03 2021

%o (Sage)

%o def A132200(n,k): return 4*binomial(n,k) - 3*bool(n==0)

%o flatten([[A132200(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 03 2021

%Y Cf. A000079, A007318, A134058, A134059, A135089.

%K nonn,tabl

%O 0,2

%A _Philippe Deléham_, Nov 19 2007