A131239 - OEIS

%I #8 Feb 26 2022 00:35:47

%S 1,1,1,1,4,1,1,5,7,1,1,8,12,10,1,1,9,24,22,13,1,1,12,33,52,35,16,1,1,

%T 13,51,85,95,51,19,1,1,16,64,148,180,156,70,22,1,1,17,88,212,348,336,

%U 238,92,25,1,1,20,105,320,560,714,574,344,117,28,1,1,21,135,425,920,1274,1330,918,477,145,31,1

%N Triangle, T(n,k) = 3*A007318(n,k) - 2*A046854(n,k), read by rows.

%C Row sums = A074878: (1, 2, 6, 14, 32, 70, 239, ...).

%C A131238 = 2*A007318 - A046854.

%H G. C. Greubel, <a href="/A131239/b131239.txt">Rows n = 0..100 of triangle, flattened</a>

%F T(n,k) = 3*A007318(n,k) - 2*A046854(n,k) as infinite lower triangular matrices.

%F T(n,k) = 3*binomial(n,k) - 2*binomial(floor((n+k)/2), k). - _G. C. Greubel_, Jul 12 2019

%e First few rows of the triangle:

%e   1;

%e   1,  1;

%e   1,  4,  1;

%e   1,  5,  7,  1;

%e   1,  8, 12, 10,  1;

%e   1,  9, 24, 22, 13,  1;

%e   1, 12, 33, 52, 35, 16, 1;

%e   ...

%t With[{B=Binomial}, Table[3*B[n,k] - 2*B[Floor[(n+k)/2], k], {n,0,12}, {k,0,n}]]//Flatten (* _G. C. Greubel_, Jul 12 2019 *)

%o (PARI) b=binomial; T(n,k) = 3*b(n,k) - 2*b((n+k)\2, k);

%o for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Jul 12 2019

%o (Magma) B:=Binomial; [3*B(n,k) - 2*B(Floor((n+k)/2), k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 12 2019

%o (Sage) b=binomial; [[3*b(n,k) - 2*b(floor((n+k)/2), k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Jul 12 2019

%o (GAP) B:=Binomial;; Flat(List([0..12], n-> List([0..n], k-> 3*B(n,k) - 2*B(Int((n+k)/2), k) ))); # _G. C. Greubel_, Jul 12 2019

%Y Cf. A007318, A046854, A131238, A074878.

%K nonn,tabl

%O 0,5

%A _Gary W. Adamson_, Jun 21 2007

%E More terms added by _G. C. Greubel_, Jul 12 2019