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.
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
Examples
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, ... ...
Links
- A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
Crossrefs
Programs
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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);
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.
Comments