|
| |
|
|
A098473
|
|
Triangle T(n,k) read by rows, T(n, k)=binomial(2*k, k)*binomial(n, k), 0<=k<=n.
|
|
5
| |
|
|
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
(list; table; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,3
|
|
|
COMMENTS
| Row sums are A026375. Diagonal sums are A026569. Principal diagonal is A000984.
|
|
|
LINKS
| 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<=k<=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 DELEHAM (kolotoko(AT)wanadoo.fr), Sep 28 2007
From Peter Bala (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..inf} 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 A117128). (End)
|
|
|
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:
|
|
|
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
| Sequence in context: A033877 A059369 A199530 * A121757 A181854 A109822
Adjacent sequences: A098470 A098471 A098472 * A098474 A098475 A098476
|
|
|
KEYWORD
| easy,nonn,tabl
|
|
|
AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Sep 09 2004
|
| |
|
|
