A117178 Riordan array (c(x^2)/sqrt(1-4*x^2), x*c(x^2)), c(x) the g.f. of A000108.

1, 0, 1, 3, 0, 1, 0, 4, 0, 1, 10, 0, 5, 0, 1, 0, 15, 0, 6, 0, 1, 35, 0, 21, 0, 7, 0, 1, 0, 56, 0, 28, 0, 8, 0, 1, 126, 0, 84, 0, 36, 0, 9, 0, 1, 0, 210, 0, 120, 0, 45, 0, 10, 0, 1, 462, 0, 330, 0, 165, 0, 55, 0, 11, 0, 1, 0, 792, 0, 495, 0, 220, 0, 66, 0, 12, 0, 1

Offset

0,4

Comments

Row sums are A058622(n+1). Diagonal sums are A001791(n+1), with interpolated zeros. Inverse is A117179.

De-aerated and rows reversed, this matrix apparently becomes A014462. The nonzero antidiagonals are embedded in several entries and apparently contain partial sums of previous nonzero antidiagonals. - Tom Copeland, May 30 2017

Links

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

Formula

T(n,k) = C(n+1, (n-k)/2) * (1 + (-1)^(n-k))/2.

Column k has e.g.f. Bessel_I(k,2x) + Bessel_I(k+2, 2x).

From G. C. Greubel, Aug 08 2022: (Start)

Sum_{k=0..n} T(n, k) = A058622(n+1).

Sum_{k=0..floor(n/2)} T(n-k, k) = ((1+(-1)^n)/2) * A001791((n+2)/2).

T(2*n, n) = ((1+(-1)^n)/2) * A052203(n/2).

T(2*n+1, n) = ((1-(-1)^n)/2) * A224274((n+1)/2).

T(2*n-1, n-1) = ((1+(-1)^n)/2) * A224274(n/2). (End)

Example

Triangle begins
    1;
    0,  1;
    3,  0,  1;
    0,  4,  0,  1;
   10,  0,  5,  0,  1;
    0, 15,  0,  6,  0,  1;
   35,  0, 21,  0,  7,  0,  1;
    0, 56,  0, 28,  0,  8,  0,  1;
  126,  0, 84,  0, 36,  0,  9,  0,  1;

Mathematica

T[n_, k_]:= Binomial[n+1, (n-k)/2]*(1+(-1)^(n-k))/2;
Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Aug 08 2022 *)

Prog

(Magma) [(1+(-1)^(n-k))*Binomial(n+1, Floor((n-k)/2))/2: k in [0..n], n in [0..15]]; // G. C. Greubel, Aug 08 2022
(SageMath)
def A117178(n, k): return (1 + (-1)^(n-k))*binomial(n+1, (n-k)//2)/2
flatten([[A117178(n, k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Aug 08 2022

Crossrefs

Cf. A001791, A014462, A052203, A058622 (row sums), A117179, A224274.

Sequence in context: A049769 A117179 A111526 * A111527 A035695 A100257

Adjacent sequences: A117175 A117176 A117177 * A117179 A117180 A117181

Keyword

easy,nonn,tabl

Author

Paul Barry, Mar 01 2006

Status

approved