A125101 T(n,k) = k*binomial(n-1,k-1) + Fibonacci(k)*binomial(n-1,k) (1 <= k <= n).

1, 2, 2, 3, 5, 3, 4, 9, 11, 4, 5, 14, 26, 19, 5, 6, 20, 50, 55, 30, 6, 7, 27, 85, 125, 105, 44, 7, 8, 35, 133, 245, 280, 182, 62, 8, 9, 44, 196, 434, 630, 560, 300, 85, 9, 10, 54, 276, 714, 1260, 1428, 1056, 477, 115, 10, 11, 65, 375, 1110, 2310, 3192, 3030, 1905, 745, 155

Offset: 1

Author

Gary W. Adamson, Nov 20 2006

Keywords

• nonn
• tabl

Comments

Row sums are s(n) = 1, 4, 11, 28, 69, 167, 400, ...

Binomial transform of the bidiagonal matrix with (1,2,3...) in the main diagonal and the Fibonacci numbers (1,1,2,3,5,8,...) in the subdiagonal.

Examples

			First few rows of the triangle:
  1;
  2,  2;
  3,  5,   3;
  4,  9,  11,   4;
  5, 14,  26,  19,   5;
  6, 20,  50,  55,  30,   6;
  7, 27,  85, 125, 105,  44,  7;
  8, 35, 133, 245, 280, 182, 62, 8;
  ...
		

Crossrefs

Cf. A000096, A051925.

Programs

Maple

with(combinat): T:=(n,k)->k*binomial(n-1,k-1)+fibonacci(k)*binomial(n-1,k): for n from 1 to 12 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form

Mathematica

Flatten[Table[k Binomial[n-1,k-1]+Fibonacci[k]Binomial[n-1,k],{n,15},{k,n}]] (* Harvey P. Dale, Nov 03 2014 *)

Formula

T(n,2) = A000096(n-1).

T(n,3) = A051925(n-1).

T(n,4) = A215862(n-3). - R. J. Mathar, Aug 10 2013

Row sums s(n) = 7*s(n-1) -17*s(n-2) +16*s(n-3) -4*s(n-4) with s(n) = A001787(n+1)/4 +A001906(n-1). - R. J. Mathar, Aug 10 2013

Extensions

Edited by N. J. A. Sloane, Nov 29 2006