A098473 Triangle T(n,k) read by rows, T(n, k) = binomial(2*k, k)*binomial(n, k), 0<=k<=n.

1, 1, 2, 1, 4, 6, 1, 6, 18, 20, 1, 8, 36, 80, 70, 1, 10, 60, 200, 350, 252, 1, 12, 90, 400, 1050, 1512, 924, 1, 14, 126, 700, 2450, 5292, 6468, 3432, 1, 16, 168, 1120, 4900, 14112, 25872, 27456, 12870, 1, 18, 216, 1680, 8820, 31752, 77616, 123552, 115830

Offset: 0

Author

Paul Barry, Sep 09 2004

Keywords

• easy
• nonn
• tabl

Comments

This sequence gives the coefficients of the Jensen polynomials (increasing powers of x) of degree n and shift 0 for the central binomial sequence A000984. For a definition of Jensen polynomials see a comment in A094436. - Wolfdieter Lang, Jun 25 2019

Examples

			Rows begin
  1;
  1,  2;
  1,  4,  6;
  1,  6, 18,  20;
  1,  8, 36,  80,  70;
  1, 10, 60, 200, 350, 252;
		

Links

• Indranil Ghosh, Rows 0..125, flattened
• O. T. Dasbach, A natural series for the natural logarithm, Electronic Journal of Combinatorics, (15) 2008 #N5.

Crossrefs

Row sums are A026375.

Antidiagonal sums are A026569.

Principal diagonal is A000984.

Cf. A002426, A081671, A098409, A098410, A094436, A126869.

Programs

Maple

A098473 := proc(n,k) binomial(2*k,k)*binomial(n,k) ; end proc:

Mathematica

Table[Binomial[2k,k]Binomial[n,k],{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Aug 15 2020 *)

PARI

T(n,k)=binomial(2*k, k)*binomial(n, k);
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print()); /* as triangle */

Formula

T(n, k) = binomial(2*k, k)*binomial(n, k).

Sum_{k=0..n} T(n,k)*x^(n-k) = A126869(n), A002426(n), A000984(n), A026375(n), A081671(n), A098409(n), A098410(n) for x = -2, -1, 0, 1, 2, 3, 4 respectively. - Philippe Deléham, Sep 28 2007

From Peter Bala, Jun 06 2011: (Start)

O.g.f.: 1/sqrt(1 - t)*1/sqrt(1 - t*(1 + 4*x)) = 1 + (2*x + 1)*t + (1 + 4*x + 6*x^2)*t^2 + ....

Let R_n(x) denote the row generating polynomials of this triangle, which begin

R_1(x) = 1 + 2*x, R_2(x) = 1 + 4*x + 6*x^2, R_3(x) = 1 + 6*x + 18*x^2 + 20*x^3.

Dasbach gives the following slowly converging series for the logarithm function:

log(x) = Sum_{n >= 1} 1/n*R_n(-1/x), valid for x >= 4.

The polynomials (1 - x)^n*R_n(x/(1 - x)) appear to be the row polynomials of A135091 (see also A171128). (End)