A131239 Triangle, T(n,k) = 3*A007318(n,k) - 2*A046854(n,k), read by rows.

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

Offset: 0

Author

Gary W. Adamson, Jun 21 2007

Keywords

• nonn
• tabl

Comments

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

A131238 = 2*A007318 - A046854.

Examples

			First few rows of the triangle:
  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;
  ...
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened

Crossrefs

Cf. A007318, A046854, A131238, A074878.

Programs

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

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

Mathematica

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 *)

PARI

b=binomial; T(n,k) = 3*b(n,k) - 2*b((n+k)\2, k);
for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Jul 12 2019

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

Formula

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

T(n,k) = 3*binomial(n,k) - 2*binomial(floor((n+k)/2), k). - G. C. Greubel, Jul 12 2019

Extensions

More terms added by G. C. Greubel, Jul 12 2019