A131067 Triangle read by rows: T(n,k) = 7*binomial(n,k) - 6 for 0 <= k <= n.

1, 1, 1, 1, 8, 1, 1, 15, 15, 1, 1, 22, 36, 22, 1, 1, 29, 64, 64, 29, 1, 1, 36, 99, 134, 99, 36, 1, 1, 43, 141, 239, 239, 141, 43, 1, 1, 50, 190, 386, 484, 386, 190, 50, 1, 1, 57, 246, 582, 876, 876, 582, 246, 57, 1, 1, 64, 309, 834, 1464, 1758, 1464, 834, 309, 64, 1

Offset: 0

Author

Gary W. Adamson, Jun 13 2007

Keywords

• nonn
• tabl

Comments

Row sums = A131068: (1, 2, 10, 32, 82, 188, 406, ...), the binomial transform of (1, 1, 7, 7, 7, ...).

Examples

			First few rows of the triangle:
  1;
  1,  1;
  1,  8,  1;
  1, 15, 15,  1;
  1, 22, 36, 22,  1;
  1, 29, 64, 64, 29, 1;
  ...
		

Links

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

Crossrefs

Cf. A131064, A131066.

Sequence m*binomial(n,k) - (m-1): A007318 (m=1), A109128 (m=2), A131060 (m=3), A131061 (m=4), A131063 (m=5), A131065 (m=6), this sequence (m=7), A131068 (m=8).

Programs

Magma

[7*Binomial(n, k) -6: k in [0..n], n in [0..10]]; // G. C. Greubel, Mar 12 2020

Maple

T := proc (n, k) if k <= n then 7*binomial(n, k)-6 else 0 end if end proc: for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # Emeric Deutsch, Jun 20 2007

Mathematica

Table[7*Binomial[n, k] -6, {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 12 2020 *)

Sage

[[7*binomial(n, k) -6 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 12 2020

Formula

G.f.: G(t,z) = (1-z-t*z+7*t*z^2)/((1-z)*(1-t*z)*(1-z-t*z)). - Emeric Deutsch, Jun 20 2007

Extensions

More terms from Emeric Deutsch, Jun 20 2007