A112519 Riordan array (1, x*c(x)*c(-x*c(x))), c(x) the g.f. of A000108.

1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 1, 4, 0, 1, 0, 12, 2, 6, 0, 1, 0, 14, 28, 3, 8, 0, 1, 0, 100, 32, 48, 4, 10, 0, 1, 0, 180, 249, 54, 72, 5, 12, 0, 1, 0, 990, 440, 455, 80, 100, 6, 14, 0, 1, 0, 2310, 2552, 792, 726, 110, 132, 7, 16, 0, 1, 0, 10920, 5876, 4836, 1248, 1070, 144, 168, 8, 18, 0, 1

Offset

0,8

Comments

Row sums are A112520. Second column is essentially A055392. Inverse is A112517. Riordan array product (1, x*c(x))*(1, x*c(-x)).

Links

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

Formula

Riordan array (1, (sqrt(3-2*sqrt(1-4*x)) - 1)/2).

T(n, k) = (k/n)*Sum_{j=0..n} (-1)^(j-k)*C(2*n-j-1, n-j)*C(2*j-k-1, j-k), with T(0, 0) = 1.

T(n, k) = (k/n)*binomial(2*n-k-1, n-1)*Hypergoemetric3F2([k-n, k/2, (1+k)/2], [k-2*n+1, k], -4), with T(0, 0) = 1. - G. C. Greubel, Jan 12 2022

Example

Triangle begins
  1;
  0,   1;
  0,   0,   1;
  0,   2,   0,   1;
  0,   1,   4,   0,  1;
  0,  12,   2,   6,  0,   1;
  0,  14,  28,   3,  8,   0,  1;
  0, 100,  32,  48,  4,  10,  0,  1;
  0, 180, 249,  54, 72,   5, 12,  0, 1;
  0, 990, 440, 455, 80, 100,  6, 14, 0, 1;

Mathematica

(* First program *)
c[x_]:= (1 - Sqrt[1-4x])/(2x);
(* The function RiordanArray is defined in A256893. *)
RiordanArray[1&, # c[#] c[-# c[#]]&, 12] // Flatten (* Jean-François Alcover, Jul 16 2019 *)
(* Second program *)
T[n_, k_]:= If[k==n, 1, (k/n)*Binomial[2*n-k-1, n-1]*HypergeometricPFQ[{k-n, k/2, (1+k)/2}, {k-2*n+1, k}, -4]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Jan 12 2022 *)

Prog

(Magma)
A112519:= func< n, k | n eq 0 and k eq 0 select 1 else (k/n)*(&+[(-1)^j*Binomial(2*n-k-j-1, n-k-j)*Binomial(2*j+k-1, j): j in [0..n-k]]) >;
[A112519(n, k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jan 12 2022
(Sage)
@CachedFunction
def A112519(n, k):
    if (k==n): return 1
    else: return (k/n)*sum( (-1)^j*binomial(2*n-k-j-1, n-k-j)*binomial(2*j+k-1, j) for j in (0..n-k) )
flatten([[A112519(n, k) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Jan 12 2022

Crossrefs

Cf. A000108, A055392, A112517, A112520.

Sequence in context: A352747 A364955 A112517 * A276981 A230305 A357293

Adjacent sequences: A112516 A112517 A112518 * A112520 A112521 A112522

Keyword

easy,nonn,tabl

Author

Paul Barry, Sep 09 2005

Status

approved