OFFSET
0,3
COMMENTS
Compare with A111528, which has a similar definition.
LINKS
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
FORMULA
Let d(n) = Product_{k = 1..n} 2*k-1 = A001147(n) denote the double factorial of odd numbers.
O.g.f. for row n: R(n,x) = ( Sum_{k >= 0} d(n+k)/d(n)*x^k )/( Sum_{k >= 0} d(n-1+k)/d(n-1)*x^k ).
R(n,x)/(1 - (2*n-1)*x*R(n,x)) = Sum_{k >= 0} d(n+k)/d(n)*x^k.
R(n,x) = 1/(1 + (2*n-1)*x - (2*n+1)*x/(1 + (2*n+1)*x - (2*n+3)*x/(1 + (2*n+3)*x - (2*n+5)*x/(1 + (2*n+5)*x - ... )))).
R(n,x) satisfies the Riccati differential equation 2*x^2*d/dx(R(n,x)) + (2*n-1)*x*R(n,x)^2 - (1 + (2*n-3)*x)*R(n,x) + 1 = 0 with R(n,0) = 1.
Applying Stokes 1982 gives A(x) = 1/(1 - 2*x/(1 - (2*n+1)*x/(1 - 4*x/(1 - (2*n+3)*x/(1 - 6*x/(1 - (2*n+5)*x/(1 - ... - 2*m*x/(1 - (2*n+2*m-1)*x/(1 - ... ))))))))), a continued fraction of Stieltjes type.
EXAMPLE
Square array begins
1, 2, 6, 26, 158, 1282, 13158, 163354, 2374078, 39456386, ...
1, 2, 10, 74, 706, 8162, 110410, 1708394, 29752066, 576037442, ...
1, 2, 14, 138, 1686, 24162, 394254, 7191018, 144786006, 3188449602, ...
1, 2, 18, 218, 3194, 53890, 1019250, 21256090, 483426010, 11895873410, ...
1, 2, 22, 314, 5326, 102722, 2197558, 51355514, 1297759918, 35208930050, ...
1, 2, 26, 426, 8178, 176802, 4206618, 108577674, 3011332338, 89141101506, ...
1, 2, 30, 554, 11846, 283042, 7396830, 208569034, 6288011206, 201404591042, ...
...
MAPLE
T := (n, k) -> coeff(series(hypergeom([n+1/2, 1], [], 2*x)/ hypergeom([n-1/2, 1], [], 2*x), x, 21), x, k):
# display as a sequence
seq(seq(T(n-k, k), k = 0..n), n = 0..10);
# display as a square array
seq(print(seq(T(n, k), k = 0..10)), n = 0..10);
CROSSREFS
KEYWORD
AUTHOR
Peter Bala, Jul 15 2022
STATUS
approved