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Sequence matrix for odd numbers.
10

%I #35 Jul 23 2014 11:00:58

%S 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,

%T 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,

%U 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

%N Sequence matrix for odd numbers.

%C Riordan array ((1+x)/(1-x)^2, x).

%C Inverse matrix is A101038.

%C Row sums yield (n+1)^2.

%C 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.

%C From _Paul Curtz_, Sep 25 2011. (Start)

%C Consider from A187870(n-2) and A171080(n)

%C 1 + 1/3 - 4/45 + 44/945 - 428/14175 =1/(1 -1/3 +1/5 -1/7 ..= Pi/4)=4/Pi.

%C For c(0)=-1, c(1)=1/3, c(2)=4/45, c(3)=44/945, c(4)=428/14175,

%C c(0)/3 + c(1)=0,

%C c(0)/5 + c(1)/3 + c(2)=0,

%C c(0)/7 + c(1)/5 + c(2)/3 + c(3)=0.

%C Hence a(n+1). Numbers are

%C -1/3 + 1/3, 1=1,

%C -1/5 + 1/9 + 4/45, 4=9-5,

%C -1/7 + 1/15 + 4/135 + 44/945 44=135-63-28. (End)

%C T(n,k) = A158405(n+1,n+1-k), 1<=k<=n. [_Reinhard Zumkeller_, Mar 31 2012]

%C From _Peter Bala_, Jul 22 2014: (Start)

%C Call this array M and for k = 0,1,2,... define M(k) to be the lower unit triangular block array

%C /I_k 0\

%C \ 0 M/

%C 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)

%H Reinhard Zumkeller, <a href="/A099375/b099375.txt">Rows n=0..150 of triangle, flattened</a>

%F Number triangle T(n, k)=if(k<=n, 2(n-k)+1, 0)=binomial(2(n-k)+1, 2(n-k))

%F 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

%e Rows start

%e 1;

%e 3,1;

%e 5,3,1;

%e 7,5,3,1;

%e 9,7,5,3,1;

%e 11,9,7,5,3,1;

%e 13,11,9,7,5,3,1;

%o (Haskell)

%o a099375 n k = a099375_row n !! k

%o a099375_row n = a099375_tabl !! n

%o a099375_tabl = iterate (\xs -> (head xs + 2) : xs) [1]

%o -- _Reinhard Zumkeller_, Mar 31 2012

%Y Cf. A005408, A004736. A208904.

%K nonn,easy,tabl

%O 0,2

%A _Paul Barry_, Jan 22 2005