OFFSET
0,8
COMMENTS
The denominators are given by 2*A130321(n,m).
The rational lower triangular matrix with entries R(n,m) = T(n,m)/(2*A130321(n,m)) = T(n,m)/2^(n-m+1) for n >= m >= 0 and 0 otherwise is the inverse of the Riordan matrix A029635.
R is the rational Riordan triangle (1/(2-x), x/(1+x)).
The numerator triangle T is the Riordan array (1/(1-x), x/(1+2*x)). From the o.g.f. of the column sequences of R and T(n,m) = 2^(n-m+1)*R(n,m).
Row sums of the rational triangle R are [1/2, seq(3/2^(n+1), for n >= 1)].
Row sums of the present triangle T give [repeat(1,2,)].
Alternating row sums of the rational triangle R give (-1)^n*A102900(n)/2^(n+1), n >= 0: 1/2, -1/4, 7/8, -25/16, 103/32, -409/64, 1639/128, -6553/256, 26215/512, ... .
Alternating row sums of the present triangle T give A084567.
The inverse of the T Riordan matrix is ((1-3*x)/(1-2*x), x/(1-2*x) = A251636.
Equals A248810 when the first column (m = 0) of ones is removed. - Georg Fischer, Jul 26 2023
LINKS
FORMULA
O.g.f. of the row polynomials P(n,x) = sum_{m=0..n} (R(n,m)*x^m of the rational triangle R: G(z,x) = sum_{n>=0} P(n,x)*z^n = (1+z)/((2-z)*(1+(1-x)*z).
O.g.f. column m of the rational triangle R: (1/(2-x))*(x/(1+x))^m, m >= 0 (Riordan property of R).
O.g.f. column m of the numerator triangle T: (1/(1-x))*(x/(1+2*x))^m, m >= 0. (Riordan property of T).
T(n, k) = k!*S(n, k) where S(n, k) is recursively defined by:
if k = 0 then 1 else if k > n then 0 else S(n-1, k-1)/k - 2*S(n-1, k). - Peter Luschny, Jan 19 2020
EXAMPLE
The triangle T(n,m) begins:
n\m 0 1 2 3 4 5 6 7 8 9 10 ...
0: 1
1: 1 1
2: 1 -1 1
3: 1 3 -3 1
4: 1 -5 9 -5 1
5: 1 11 -23 19 -7 1
6: 1 -21 57 -61 33 -9 1
7: 1 43 -135 179 -127 51 -11 1
8: 1 -85 313 -493 433 -229 73 -13 1
9: 1 171 -711 1299 -1359 891 -375 99 -15 1
...
The rational Riordan triangle R(n,m) begins:
n\m 0 1 2 3 4 5 ...
0: 1/2
1: 1/4 1/2
2: 1/8 -1/4 1/2
3: 1/16 3/8 -3/4 1/2
4: 1/32 -5/16 9/8 -5/4 1/2
5: 1/64 11/3 -23/1 19/8 -7/4 1/2
...
For more rows see the link.
MAPLE
A251634 := proc(n, k) local S; S := proc(n, k) option remember; `if`(k = 0, 1,
`if`(k > n, 0, S(n-1, k-1)/k - 2*S(n-1, k))) end: k!*S(n, k) end:
seq(seq(A251634(n, k), k=0..n)), n=0..9); # Peter Luschny, Jan 19 2020
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Jan 09 2015
STATUS
approved