A131238 Triangle read by rows: T(n,k) = 2*binomial(n,k) - binomial(floor((n+k)/2), k) (0 <= k <= n).

1, 1, 1, 1, 3, 1, 1, 4, 5, 1, 1, 6, 9, 7, 1, 1, 7, 17, 16, 9, 1, 1, 9, 24, 36, 25, 11, 1, 1, 10, 36, 60, 65, 36, 13, 1, 1, 12, 46, 102, 125, 106, 49, 15, 1, 1, 13, 62, 148, 237, 231, 161, 64, 17, 1, 1, 15, 75, 220, 385, 483, 392, 232, 81, 19, 1, 1, 16, 95, 295, 625, 868, 896, 624, 321, 100, 21, 1

Offset

0,5

Comments

Row sums = A027934: (1, 2, 5, 11, 24, 51, 107, ...).

A131239 = 3*A007318 - 2*A046854.

Links

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

Formula

T(n,k) = 2*A007318(n,k) - A046854(n,k) as infinite lower triangular matrices, where A007318 = Pascal's triangle and A046854 = Pascal's triangle with repeats, by columns.

Example

First few rows of the triangle:
  1;
  1,  1;
  1,  3,  1;
  1,  4,  5,  1;
  1,  6,  9,  7,  1;
  1,  7, 17, 16,  9,  1;
  1,  9, 24, 36, 25, 11,  1;
  1, 10, 36, 60, 65, 36, 13, 1;
  ...

Maple

T := proc (n, k) options operator, arrow; 2*binomial(n, k)-binomial(floor((1/2)*n+(1/2)*k), k) end proc: for n from 0 to 9 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form - Emeric Deutsch, Jul 09 2007

Mathematica

With[{B = Binomial}, Table[2*B[n, k] - 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) = 2*b(n, k) - 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; [2*B(n, k) - B(Floor((n+k)/2), k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 12 2019
(Sage) b=binomial; [[2*b(n, k) - 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-> 2*B(n, k) - B(Int((n+k)/2), k) ))); # G. C. Greubel, Jul 12 2019

Crossrefs

Cf. A027934, A131239, A007318, A046854.

Sequence in context: A249488 A275204 A321876 * A133380 A343168 A105687

Adjacent sequences: A131235 A131236 A131237 * A131239 A131240 A131241

Keyword

nonn,tabl

Author

Gary W. Adamson, Jun 21 2007

Extensions

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

Status

approved