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

0,3

Comments

Compare with A111528 and A355721, which have similar definitions and properties.

Links

Table of n, a(n) for n=0..51.

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

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.

Example

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, ...

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);

Crossrefs

Cf. A112936 (row 0), A355794 (row 1), A355795 (row 2), A355796 (row 3), A355797 (row 4). Cf. A008544, A111528, A355721.

Sequence in context: A327149 A351372 A356411 * A173424 A143081 A179658

Adjacent sequences: A355790 A355791 A355792 * A355794 A355795 A355796

Keyword

nonn,tabl,easy

Author

Peter Bala, Jul 17 2022

Status

approved