A124305 Riordan array (1, 2*sqrt(3)*sin(arcsin(3*sqrt(3)*x/2)/3)/3).

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 3, 0, 3, 0, 1, 0, 0, 7, 0, 4, 0, 1, 0, 12, 0, 12, 0, 5, 0, 1, 0, 0, 30, 0, 18, 0, 6, 0, 1, 0, 55, 0, 55, 0, 25, 0, 7, 0, 1, 0, 0, 143, 0, 88, 0, 33, 0, 8, 0, 1

Offset

0,13

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

Sum_{k=0..n} T(n, k) = A047749(n).

Sum_{k=0..floor(n/2)} T(n-k, k) = (1/2)*(1 + (-1)^n)*A098746(n/2).

From G. C. Greubel, Aug 19 2023: (Start)

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

T(n, n) = 1.

T(n, n-2) = A001477(n-2).

T(n, n-4) = A055998(n-4).

T(n, n-6) = A111396(n-6).

T(n, 0) = 0^n.

T(n, 1) = ((1-(-1)^n)/2)*A001764(floor((n-1)/2)).

T(n, 2) = ((1+(-1)^n)/2)*A006013(floor((n-2)/2)).

Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n * A047749(n). (End)

Example

Triangle begins
  1,
  0,  1,
  0,  0,  1,
  0,  1,  0,  1,
  0,  0,  2,  0,  1,
  0,  3,  0,  3,  0,  1,
  0,  0,  7,  0,  4,  0,  1,
  0, 12,  0, 12,  0,  5,  0,  1
From Paul Barry, Sep 28 2009: (Start)
Production matrix is
  0, 1,
  0, 0, 1,
  0, 1, 0, 1,
  0, 0, 1, 0, 1,
  0, 1, 0, 1, 0, 1,
  0, 0, 1, 0, 1, 0, 1,
  0, 1, 0, 1, 0, 1, 0, 1,
  0, 0, 1, 0, 1, 0, 1, 0, 1,
  0, 1, 0, 1, 0, 1, 0, 1, 0, 1,
  0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1 (End)

Mathematica

A124305[n_, k_]:= If[n==0, 1, (1/2)*(1+(-1)^(n-k))*(k/n)*Binomial[n +(n-k)/2 -1, (n-k)/2]];
Table[A124305[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Aug 19 2023 *)

Prog

(Magma)
A124305:= func< n, k | n eq 0 select 1 else (1/2)*(1+(-1)^(n-k))*(k/n)*Binomial(n + Floor((n-k)/2) -1, n-1) >;
[A124305(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 25 2023
(SageMath)
def A124305(n, k): return 1 if n==0 else ((n-k+1)%2)*k*binomial(n + (n-k)//2 -1, n-1)//n
flatten([[A124305(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Aug 25 2023

Crossrefs

Cf. A001477, A001764, A006013, A055998.

Cf. A047749 (row sums), A098746 (diagonal sums), A124304 (inverse).

Sequence in context: A052173 A177825 A175790 * A349395 A087073 A297173

Adjacent sequences: A124302 A124303 A124304 * A124306 A124307 A124308

Keyword

easy,nonn,tabl

Author

Paul Barry, Oct 25 2006

Status

approved