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A355721
Square table, read by antidiagonals: the g.f. for row n is given recursively by (2*n-1)*x*R(n,x) = 1 + (2*n-3)*x - 1/R(n-1,x) for n >= 1 with the initial value R(0,x) = Sum_{k >= 0} A112934(k+1)*x^k.
9
1, 1, 2, 1, 2, 6, 1, 2, 10, 26, 1, 2, 14, 74, 158, 1, 2, 18, 138, 706, 1282, 1, 2, 22, 218, 1686, 8162, 13158, 1, 2, 26, 314, 3194, 24162, 110410, 163354, 1, 2, 30, 426, 5326, 53890, 394254, 1708394, 2374078, 1, 2, 34, 554, 8178, 102722, 1019250, 7191018, 29752066, 39456386
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.
Row 0: A112934(n+1); Row 1; A000698(n+1).
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
Cf. A112934 (row 0), A000698 (row 1), A355722 (row 2), A355723 (row 3), A355724 (row 4), A355725 (row 5). Cf. A001147, A111528.
Sequence in context: A133643 A008305 A208763 * A249027 A307584 A266183
KEYWORD
nonn,tabl,easy
AUTHOR
Peter Bala, Jul 15 2022
STATUS
approved