OFFSET
0,2
COMMENTS
Riordan array ((1+x)/(1-x)^2, x).
Inverse matrix is A101038.
Row sums yield (n+1)^2.
Diagonal sums yield sum{k=0..floor(n/2),2(n-2k)+1}=C(n+2,2)=A000217(n+1). Note that sum{k=0..n,2(n-2k)+1}=n+1.
From Paul Curtz, Sep 25 2011. (Start)
1 + 1/3 - 4/45 + 44/945 - 428/14175 =1/(1 -1/3 +1/5 -1/7 ..= Pi/4)=4/Pi.
For c(0)=-1, c(1)=1/3, c(2)=4/45, c(3)=44/945, c(4)=428/14175,
c(0)/3 + c(1)=0,
c(0)/5 + c(1)/3 + c(2)=0,
c(0)/7 + c(1)/5 + c(2)/3 + c(3)=0.
Hence a(n+1). Numbers are
-1/3 + 1/3, 1=1,
-1/5 + 1/9 + 4/45, 4=9-5,
-1/7 + 1/15 + 4/135 + 44/945 44=135-63-28. (End)
T(n,k) = A158405(n+1,n+1-k), 1<=k<=n. [Reinhard Zumkeller, Mar 31 2012]
From Peter Bala, Jul 22 2014: (Start)
Call this array M and for k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0 M/
having the k x k identity matrix I_k as the upper left block; in particular, M(0) = M. The infinite matrix product M(0)*M(1)*M(2)*..., which is clearly well-defined, is equal to A208904. (End)
LINKS
Reinhard Zumkeller, Rows n=0..150 of triangle, flattened
FORMULA
Number triangle T(n, k)=if(k<=n, 2(n-k)+1, 0)=binomial(2(n-k)+1, 2(n-k))
a(n)=2*A004736(n)-1; a(n)=2*((t*t+3*t+4)/2-n)-1, where t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Feb 08 2013
EXAMPLE
Rows start
1;
3,1;
5,3,1;
7,5,3,1;
9,7,5,3,1;
11,9,7,5,3,1;
13,11,9,7,5,3,1;
PROG
(Haskell)
a099375 n k = a099375_row n !! k
a099375_row n = a099375_tabl !! n
a099375_tabl = iterate (\xs -> (head xs + 2) : xs) [1]
-- Reinhard Zumkeller, Mar 31 2012
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Jan 22 2005
STATUS
approved