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

0,5

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

Links

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

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

Example

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;

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 *)

Prog

(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
(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

Crossrefs

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

Cf. A000302, A132045.

Sequence in context: A356113 A188461 A188474 * A173043 A082046 A132787

Adjacent sequences: A173043 A173044 A173045 * A173047 A173048 A173049

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 08 2010

Extensions

Edited by G. C. Greubel, Feb 16 2021

Status

approved