A099375 Sequence matrix for odd numbers.

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

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

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

Cf. A005408, A004736. A208904.

Sequence in context: A360325 A084305 A347066 * A348835 A130301 A133601

Adjacent sequences: A099372 A099373 A099374 * A099376 A099377 A099378

Keyword

nonn,easy,tabl

Author

Paul Barry, Jan 22 2005

Status

approved