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

0,5

Comments

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

A131238 = 2*A007318 - A046854.

Links

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

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

Example

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;
  ...

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 *)

Prog

(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
(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
(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
(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

Crossrefs

Cf. A007318, A046854, A131238, A074878.

Sequence in context: A171142 A174037 A173077 * A114033 A334426 A209417

Adjacent sequences: A131236 A131237 A131238 * A131240 A131241 A131242

Keyword

nonn,tabl

Author

Gary W. Adamson, Jun 21 2007

Extensions

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

Status

approved