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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; text; internal format)
OFFSET

0,3

COMMENTS

Row sums are A026375. Diagonal sums are A026569. Principal diagonal is A000984.

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.

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

Cf. A000984, A026375, A026569.

Sequence in context: A199530 A208765 A232335 * A121757 A219441 A219142

Adjacent sequences:  A098470 A098471 A098472 * A098474 A098475 A098476

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, Sep 09 2004

STATUS

approved

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Last modified May 23 16:31 EDT 2017. Contains 286925 sequences.