A114123 Riordan array (1/(1-x), x*(1+x)^2/(1-x)^2).

1, 1, 1, 1, 5, 1, 1, 13, 9, 1, 1, 25, 41, 13, 1, 1, 41, 129, 85, 17, 1, 1, 61, 321, 377, 145, 21, 1, 1, 85, 681, 1289, 833, 221, 25, 1, 1, 113, 1289, 3653, 3649, 1561, 313, 29, 1, 1, 145, 2241, 8989, 13073, 8361, 2625, 421, 33, 1, 1, 181, 3649, 19825, 40081, 36365, 16641, 4089, 545, 37, 1

Offset: 0

Author

Paul Barry, Feb 07 2006, Oct 22 2006

Keywords

• easy
• nonn
• tabl

Comments

Row sums are A099463(n+1). Diagonal sums are A116404.

Triangle formed of even-numbered columns of the Delannoy triangle A008288. - Philippe Deléham, Mar 11 2013

Examples

			Triangle begins
  1;
  1,  1;
  1,  5,   1;
  1, 13,   9,   1;
  1, 25,  41,  13,   1;
  1, 41, 129,  85,  17,  1;
  1, 61, 321, 377, 145, 21, 1;
		

Links

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

Crossrefs

Cf. A008288, A099463 (row sums), A116404 (diagonal sums), A184883.

Programs

Magma

T:= func< n, k | (&+[Binomial(2*k, j)*Binomial(n-k, j)*2^j: j in [0..n-k]]) >;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2021

Maple

T := (n,k) -> hypergeom([-2*k, k-n], [1], 2);
seq(seq(round(evalf(T(n,k),99)),k=0..n),n=0..9); # Peter Luschny, Sep 16 2014

Mathematica

T[n_, k_] := Hypergeometric2F1[-2k, k-n, 1, 2];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* Jean-François Alcover, Jun 13 2019 *)

Sage

def A114123(n,k): return round( hypergeometric([-2*k, k-n], [1], 2) )
flatten([[A114123(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Nov 20 2021

Formula

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

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

Sum_{k=0..n} T(n, k) = A099463(n+1).

Sum_{k=0..floor(n/2)} T(n, k) = A116404(n).

T(n, k) = hypergeom([-2*k, k-n], [1], 2). - Peter Luschny, Sep 16 2014

T(n, n-k) = A184883(n, k). - G. C. Greubel, Nov 20 2021