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

Author

Paul Barry, Mar 01 2006

Keywords

• easy
• nonn
• tabl

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

Examples

			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;
		

Links

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

Crossrefs

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

Programs

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

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 *)

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

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)