A131061 Triangle read by rows: T(n,k) = 4*binomial(n,k) - 3 for 0 <= k <= n.

1, 1, 1, 1, 5, 1, 1, 9, 9, 1, 1, 13, 21, 13, 1, 1, 17, 37, 37, 17, 1, 1, 21, 57, 77, 57, 21, 1, 1, 25, 81, 137, 137, 81, 25, 1, 1, 29, 109, 221, 277, 221, 109, 29, 1, 1, 33, 141, 333, 501, 501, 333, 141, 33, 1, 1, 37, 177, 477, 837, 1005, 837, 477, 177, 37, 1

Offset

0,5

Comments

Row sums = A131062: (1, 2, 7, 20, 49, 110, 235, ...); the binomial transform of (1, 1, 4, 4, 4, ...).

Triangle equals 4*A007318 - 3*A000012 as infinite lower triangular matrices. - Emeric Deutsch, Jun 21 2007

Links

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

Formula

G.f.:(1 - z - t*z + 4*t*z^2)/((1-z)*(1-t*z)*(1-z-t*z)). - Emeric Deutsch, Jun 21 2007

Example

First few rows of the triangle are
  1;
  1,  1;
  1,  5,  1;
  1,  9,  9,  1;
  1, 13, 21, 13,  1;
  1, 17, 37, 37, 17,  1;
  1, 21, 57, 77, 57, 21, 1;
  ...

Maple

T := proc (n, k) if k <= n then 4*binomial(n, k)-3 else 0 end if end proc; for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form - Emeric Deutsch, Jun 21 2007

Mathematica

Table[4*Binomial[n, k] -3, {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 12 2020 *)

Prog

(Magma) [4*Binomial(n, k) -3: k in [0..n], n in [0..10]]; // G. C. Greubel, Mar 12 2020
(Sage) [[4*binomial(n, k) -3 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 12 2020

Crossrefs

Cf. A109128, A123203, A131060, A131063, A131064, A131065, A131066, A131067, A131068.

Sequence in context: A157174 A183450 A296128 * A157169 A081578 A184883

Adjacent sequences: A131058 A131059 A131060 * A131062 A131063 A131064

Keyword

nonn,tabl

Author

Gary W. Adamson, Jun 13 2007

Extensions

More terms from Emeric Deutsch, Jun 21 2007

Status

approved