A187870 -id:A187870 - cp's OEIS frontend

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

Paul Barry, Jan 22 2005

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)

			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;

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened

Cf. A005408, A004736. A208904.

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

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

A195466 Denominator of the coefficient of x^(2n) in expansion of 1/x^4 - 1/(3x^2) - 1/(x^3arctanh(x)).

Original entry on oeis.org

45, 945, 14175, 467775, 638512875, 1915538625, 488462349375, 7795859096025, 32157918771103125, 316985199315159375, 3028793579456347828125, 478230565177318078125, 3952575621190533915703125, 28304394023345413370350078125, 7217620475953080409439269921875, 21652861427859241228317809765625

Offset: 0

R. J. Mathar, Sep 21 2011

Prepending 3 to the data gives the denominators of the odd powers in the expansion of 1/arctan(x). - Peter Luschny, Oct 04 2014

Cf. A187870 (numerator).

A195466 := proc(n)
        1/x^4 -1/(3*x^2) -1/(x^3*arctanh(x)) ;
        coeftayl(%,x=0,2*n) ;
        denom(%) ;
end proc
seq(A195466(n),n=0..15) ;
# Or
seq(denom(coeff(series(1/arctan(x),x,2*n+2),x,2*n+1)),n=1..16); # Peter Luschny, Oct 04 2014

a[n_] := Sum[(2^(j+1)*Binomial[2*n+3, j]*Sum[(k!*StirlingS1[j+k, j]*StirlingS2[j+1, k])/(j+k)!, {k, 0, j+1}])/(j+1), {j, 0, 2*n+3}]/(2*n+3); Table[a[n] // Denominator, {n, 0, 15}] (* Jean-François Alcover, Jul 03 2013, after Vladimir Kruchinin's formula in A216272 *)

1/x^4 - 1/(3x^2) - 1/(x^3*arctanh x) = 4/45 + 44*x^2/945 + 428*x^4/14175 + 10196*x^6/467775 + ...

cp's OEIS Frontend

A099375 Sequence matrix for odd numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Formula

A195466 Denominator of the coefficient of x^(2n) in expansion of 1/x^4 - 1/(3x^2) - 1/(x^3arctanh(x)).

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

Formula

cp's OEIS Frontend

A099375 Sequence matrix for odd numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Formula

A195466 Denominator of the coefficient of x^(2n) in expansion of 1/x^4 - 1/(3*x^2) - 1/(x^3*arctanh(x)).

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

Formula

A195466 Denominator of the coefficient of x^(2n) in expansion of 1/x^4 - 1/(3x^2) - 1/(x^3arctanh(x)).