cp's OEIS Frontend

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.

%I A355721 #13 Jul 18 2022 19:47:17
%S A355721 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,
%T A355721 1686,8162,13158,1,2,26,314,3194,24162,110410,163354,1,2,30,426,5326,
%U A355721 53890,394254,1708394,2374078,1,2,34,554,8178,102722,1019250,7191018,29752066,39456386
%N 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.
%C A355721 Compare with A111528, which has a similar definition.
%H A355721 A. N. Stokes, <a href="https://doi.org/10.1017/S0004972700005219">Continued fraction solutions of the Riccati equation</a>, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
%F A355721 Let d(n) = Product_{k = 1..n} 2*k-1 = A001147(n) denote the double factorial of odd numbers.
%F A355721 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 ).
%F A355721 R(n,x)/(1 - (2*n-1)*x*R(n,x)) = Sum_{k >= 0} d(n+k)/d(n)*x^k.
%F A355721 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 - ... )))).
%F A355721 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.
%F A355721 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.
%F A355721 Row 0: A112934(n+1); Row 1; A000698(n+1).
%e A355721 Square array begins
%e A355721 1, 2,  6,  26,   158,    1282,   13158,    163354,    2374078,     39456386, ...
%e A355721 1, 2, 10,  74,   706,    8162,  110410,   1708394,   29752066,    576037442, ...
%e A355721 1, 2, 14, 138,  1686,   24162,  394254,   7191018,  144786006,   3188449602, ...
%e A355721 1, 2, 18, 218,  3194,   53890, 1019250,  21256090,  483426010,  11895873410, ...
%e A355721 1, 2, 22, 314,  5326,  102722, 2197558,  51355514, 1297759918,  35208930050, ...
%e A355721 1, 2, 26, 426,  8178,  176802, 4206618, 108577674, 3011332338,  89141101506, ...
%e A355721 1, 2, 30, 554, 11846,  283042, 7396830, 208569034, 6288011206, 201404591042, ...
%e A355721 ...
%p A355721 T := (n,k) -> coeff(series(hypergeom([n+1/2, 1], [], 2*x)/ hypergeom([n-1/2, 1], [], 2*x), x, 21), x, k):
%p A355721 # display as a sequence
%p A355721 seq(seq(T(n-k,k), k = 0..n), n = 0..10);
%p A355721 # display as a square array
%p A355721 seq(print(seq(T(n,k), k = 0..10)), n = 0..10);
%Y A355721 Cf. A112934 (row 0), A000698 (row 1), A355722 (row 2), A355723 (row 3), A355724 (row 4), A355725 (row 5). Cf. A001147, A111528.
%K A355721 nonn,tabl,easy
%O A355721 0,3
%A A355721 _Peter Bala_, Jul 15 2022