A140751 Triangle read by rows, X^n * [1,0,0,0,...] where X = an infinite tridiagonal matrix with (1,0,1,0,1,...) in the main and subdiagonals and (1,1,1,...) in the subsubdiagonal.

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 3, 3, 3, 3, 1, 1, 1, 4, 4, 6, 6, 4, 4, 1, 1, 1, 5, 5, 10, 10, 10, 10, 5, 5, 1, 1, 1, 6, 6, 15, 15, 20, 20, 15, 15, 6, 6, 1, 1, 1, 7, 7, 21, 21, 35, 35, 35, 35, 21, 21, 7, 7, 1

Offset

0,7

Links

G. C. Greubel, Rows n = 0..50, flattened

Formula

From G. C. Greubel, Oct 23 2023: (Start)

Sum_{k=0..n} T(n, k) = A000079(n).

Sum_{k=0..2*n-1} T(n, k) = A000918(n+1), n >= 1.

Sum_{k=0..2*n} (-1)^k*T(n, k) = 1. (End)

Example

First few rows of the triangle are;
  1;
  1, 1, 1;
  1, 1, 2, 2,  1;
  1, 1, 3, 3,  3,  3,  1;
  1, 1, 4, 4,  6,  6,  4,  4,  1;
  1, 1, 5, 5, 10, 10, 10, 10,  5,  5,  1;
  1, 1, 6, 6, 15, 15, 20, 20, 15, 15,  6,  6, 1;
  1, 1, 7, 7, 21, 21, 35, 35, 35, 35, 21, 21, 7, 7, 1;
  ...

Mathematica

row[n_]:= Append[Table[Binomial[n, k], {k, 0, n-1}, {2}], 1]//Flatten;
Table[row[n], {n, 0, 7}]//Flatten (* Jean-François Alcover, Aug 02 2019 *)

Prog

(Sage)
@CachedFunction
def T(n, k):  # Triangle in centered form.
    if abs(k) > n: return 0
    if n == k: return 1
    even = lambda n: 1 if 2.divides(n) else 0
    odd  = lambda n: 1 if 2.divides(n+1) else 0
    return T(n-1, k-1) + odd(n-k)*T(n-1, k) + even(n-k)*T(n-1, k+1)
for n in (0..7): [T(n, k) for k in (-n..n)]  # Peter Luschny, Nov 22 2013
(Magma)
A140751:=func< n, k | k mod 2 eq 0 select Binomial(n, Floor(k/2)) else k mod 2 eq 1 select Binomial(n, Floor((k-1)/2)) else 0 >;
[A140751(n, k): k in [0..2*n], n in [0..12]]; // G. C. Greubel, Oct 23 2023

Crossrefs

Cf. A000079, A000918, A007318 (Pascal's triangle), A140750.

Cf. A000225 (row sums), A001405 (central terms).

Sequence in context: A291120 A025485 A219365 * A259922 A162741 A340145

Adjacent sequences: A140748 A140749 A140750 * A140752 A140753 A140754

Keyword

nonn,tabf

Author

Gary W. Adamson & Roger L. Bagula, May 26 2008

Status

approved