A168625 Triangle T(n,k) = 8*binomial(n,k) - 7 with columns 0 <= k <= n.

1, 1, 1, 1, 9, 1, 1, 17, 17, 1, 1, 25, 41, 25, 1, 1, 33, 73, 73, 33, 1, 1, 41, 113, 153, 113, 41, 1, 1, 49, 161, 273, 273, 161, 49, 1, 1, 57, 217, 441, 553, 441, 217, 57, 1, 1, 65, 281, 665, 1001, 1001, 665, 281, 65, 1, 1, 73, 353, 953, 1673, 2009, 1673, 953, 353, 73, 1

Offset: 0

Author

Roger L. Bagula, Dec 01 2009

Keywords

• nonn
• easy
• tabl

Comments

Triangle T(n,k): the coefficient [x^k] of the polynomial 8*(x+1)^n -7*( x^(n+1) - 1)/(x-1).

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  9,   1;
  1, 17,  17,   1;
  1, 25,  41,  25,    1;
  1, 33,  73,  73,   33,    1;
  1, 41, 113, 153,  113,   41,    1;
  1, 49, 161, 273,  273,  161,   49,   1;
  1, 57, 217, 441,  553,  441,  217,  57,   1;
  1, 65, 281, 665, 1001, 1001,  665, 281,  65,  1;
  1, 73, 353, 953, 1673, 2009, 1673, 953, 353, 73, 1;
		

Links

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

Crossrefs

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), A131067 (m=7), this sequence (m=8).

Programs

Magma

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

Maple

A168625:= (n,k) -> 8*binomial(n, k) -7; seq(seq(A168625(n, k), k = 0..n), n = 0.. 10); # G. C. Greubel, Mar 12 2020

Mathematica

m = 8; p[x_, n_]:= FullSimplify[ExpandAll[m*(x+1)^n -(m-1)(x^(n+1) -1)/(x-1)]];
Table[CoefficientList[p[x, n], x], {n,0,10}]//Flatten
Table[8*Binomial[n, k] -7, {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 12 2020 *)

Sage

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

Formula

T(n,k) = [x^k] ( 8*(x+1)^n-7*Sum_{s=0..n} x^s ) = 8*A007318(n,k) - 7. - R. J. Mathar, Sep 02 2011

Extensions

Definition simplified by R. J. Mathar, Sep 02 2011