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A099375 Sequence matrix for odd numbers. 10
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, 15, 13, 11, 9, 7, 5, 3, 1, 17, 15, 13, 11, 9, 7, 5, 3, 1, 19, 17, 15, 13, 11, 9, 7, 5, 3, 1, 21, 19, 17, 15, 13, 11, 9, 7, 5, 3, 1, 23, 21, 19, 17, 15, 13, 11, 9, 7, 5, 3, 1, 25, 23, 21, 19, 17, 15, 13, 11, 9 (list; table; graph; refs; listen; history; text; internal format)
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)
Consider from A187870(n-2) and A171080(n)
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
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
Sequence in context: A360325 A084305 A347066 * A348835 A130301 A133601
KEYWORD
nonn,easy,tabl
AUTHOR
Paul Barry, Jan 22 2005
STATUS
approved

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Last modified June 24 15:31 EDT 2024. Contains 373679 sequences. (Running on oeis4.)