A173046 Triangle T(n, k, q) = binomial(n, k) + q^n*binomial(n-2, k-1) - 1 with T(n, 0) = T(n, n) = 1 and q = 2, read by rows.

1, 1, 1, 1, 5, 1, 1, 10, 10, 1, 1, 19, 37, 19, 1, 1, 36, 105, 105, 36, 1, 1, 69, 270, 403, 270, 69, 1, 1, 134, 660, 1314, 1314, 660, 134, 1, 1, 263, 1563, 3895, 5189, 3895, 1563, 263, 1, 1, 520, 3619, 10835, 18045, 18045, 10835, 3619, 520, 1, 1, 1033, 8236, 28791, 57553, 71931, 57553, 28791, 8236, 1033, 1

Offset: 0

Author

Roger L. Bagula, Feb 08 2010

Keywords

• nonn
• tabl

Comments

The triangle sequences having the form T(n,k,q) = binomial(n, k) + q^n*binomial(n-2, k-1) - 1 have the row sums Sum_{k=0..n} T(n,k,q) = 2^(n-2)*q^n + 2^n - (n-1) - (5/4)*[n=0] -(q/2)*[n=1]. - G. C. Greubel, Feb 16 2021

Examples

			Triangle begins as:
  1;
  1,    1;
  1,    5,    1;
  1,   10,   10,     1;
  1,   19,   37,    19,     1;
  1,   36,  105,   105,    36,     1;
  1,   69,  270,   403,   270,    69,     1;
  1,  134,  660,  1314,  1314,   660,   134,     1;
  1,  263, 1563,  3895,  5189,  3895,  1563,   263,    1;
  1,  520, 3619, 10835, 18045, 18045, 10835,  3619,  520,    1;
  1, 1033, 8236, 28791, 57553, 71931, 57553, 28791, 8236, 1033, 1;
		

Links

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

Crossrefs

Cf. A132044 (q=0), A173075 (q=1), this sequence (q=2), A173047 (q=3).

Cf. A000302, A132045.

Programs

Magma

T:= func< n,k,q | k eq 0 or k eq n select 1 else Binomial(n,k) + q^n*Binomial(n-2,k-1) -1 >;
[T(n,k,2): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 16 2021

Mathematica

T[n_, m_, q_]:= If[k==0 || k==n, 1, Binomial[n, k] +(q^n)*Binomial[n-2, k-1] -1];
Table[T[n,k,2], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Feb 16 2021 *)

Sage

def T(n,k,q): return 1 if (k==0 or k==n) else binomial(n,k) + q^n*binomial(n-2,k-1) -1
flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 16 2021

Formula

T(n, k, q) = binomial(n, k) + q^n*binomial(n-2, k-1) - 1 with T(n, 0) = T(n, n) = 1 and q = 2.

Sum_{k=0..n} T(n, k, 2) = 4^(n-1) + 2^n - (n-1) - (5/4)*[n=0] = A000302(n-1) + A132045(n) - (5/4)*[n=0]. - [n=1]. - G. C. Greubel, Feb 16 2021

Extensions

Edited by G. C. Greubel, Feb 16 2021