A355793 Square table, read by antidiagonals: the g.f. for row n is given recursively by (3*n-1)*x*R(n,x) = 1 + (3*n-4)*x - 1/R(n-1,x) for n >= 1 with the initial value R(0,x) = Sum_{k >= 0} A112936(k+1)*x^k.
1, 1, 3, 1, 3, 15, 1, 3, 24, 111, 1, 3, 33, 282, 1131, 1, 3, 42, 507, 4236, 14943, 1, 3, 51, 786, 9609, 76548, 243915, 1, 3, 60, 1119, 17736, 212835, 1608864, 4742391, 1, 3, 69, 1506, 29103, 459768, 5350785, 38488152, 106912131, 1, 3, 78, 1947, 44196, 859143, 13333488
Offset: 0
Examples
Square array begins 1, 3, 15, 111, 1131, 14943, 243915, 4742391, 106912131, ... 1, 3, 24, 282, 4236, 76548, 1608864, 38488152, 1032125136, ... 1, 3, 33, 507, 9609, 212835, 5350785, 149961675, 4628365305, ... 1, 3, 42, 786, 17736, 459768, 13333488, 425600976, 14791250688, ... 1, 3, 51, 1119, 29103, 859143, 28091463, 1002057591, 38606468343, ... 1, 3, 60, 1506, 44196, 1458588, 52917360, 2080630776, 87823112496, ... 1, 3, 69, 1947, 63501, 2311563, 91949469, 3943276347, 180679742061, ... 1, 3, 78, 2442, 87504, 3477360, 150259200, 6970190160, 344116224960, ...
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+2/3, 1], [], 3*x)/ hypergeom([n-1/3, 1], [], 3*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 t(n) = Product_{k = 1..n} 3*k-1 = A008544(n) (triple factorial numbers).
O.g.f. for row n >= 0: R(n,x) = ( Sum_{k >= 0} t(n+k)/t(n)*x^k )/( Sum_{k >= 0} t(n-1+k)/t(n-1)*x^k ).
R(n,x)/(1 - (3*n-1)*x*R(n,x)) = Sum_{k >= 0} t(n+k)/t(n)*x^k.
R(n,x) = 1/(1 + (3*n-1)*x - (3*n+2)*x/(1 + (3*n+2)*x - (3*n+5)*x/(1 + (3*n+5)*x - (3*n+8)*x/(1 + (3*n+8)*x - ... )))) (continued fraction).
R(n,x) satisfies the Riccati differential equation 3*x^2*d/dx(R(n,x)) + (3*n-1)*x*R(n,x)^2 - (1 + (3*n-4)*x)*R(n,x) + 1 = 0 with R(n,0) = 1.
Applying Stokes 1982 gives R(n,x) = 1/(1 - 3*x/(1 - (3*n+2)*x/(1 - 6*x/(1 - (3*n+5)*x/(1 - 9*x/(1 - (3*n+8)*x/(1 - 12*x/(1 - ...)))))))), a continued fraction of Stieltjes type.
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