OFFSET
0,3
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
LINKS
Indranil Ghosh, Rows 0..125, flattened
O. T. Dasbach, A natural series for the natural logarithm, Electronic Journal of Combinatorics, (15) 2008 #N5.
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.
EXAMPLE
Rows begin
1;
1, 2;
1, 4, 6;
1, 6, 18, 20;
1, 8, 36, 80, 70;
1, 10, 60, 200, 350, 252;
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 *)
PROG
(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 */
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Sep 09 2004
STATUS
approved