%I #25 Nov 17 2024 07:36:32

%S 1,1,1,1,1,2,1,1,2,5,1,1,2,6,14,1,1,2,7,21,42,1,1,2,8,30,78,132,1,1,2,

%T 9,41,136,299,429,1,1,2,10,54,222,630,1172,1430,1,1,2,11,69,342,1221,

%U 2959,4677,4862,1,1,2,12,86,502,2192,6774,14058,18947,16796,1,1,2,13,105,708,3687,14129,37853,67472,77746,58786,1,1,2,14,126,966,5874,27184

%N Square array T(n, k) read by rising antidiagonals. Row n has the ordinary generating function (-(n*x^3-(n+1)*x^2+x) + sqrt((n*x^3-(n+1)*x^2+x)^2 - 4*(x^3-x^2)*((n+1)*x^2-x)))/(2*(x^3-x^2)).

%C The Hankel sequence transform of row n satisfies the Somos-4 recurrence c(k) = (c(k-1) * c(k-3) + n*c(k-2)^2) / c(k-4). All Somos-4 sequences which are beginning with 1, 1, 1, 1, n, ... will be covered, but the Hankel transform will start with the terms 1, n, ... in each case.

%F The generating function A(x) of row n satisfies: 0 = (x^3 - x^2)*A(x)^2 + (n*x^3 - (n+1)*x^2 - x)*A(x) + ((n+1)*x^2 - x).

%F Let d(m, n) = ( d(m-3, n)*d(m-2, n) + n)/( d(m-5, n)*d(m-4, n)*d(m-3, n)^2*d(m-2, n)^2*d(m-1, n) ) for m = even and d(m, n) = 1/( d(m-1, n)*d(m-2, n) ) for m = odd with d( < 1 , n) = 1, then the generating function of row n can be expanded as continued fractions: 1/(1 - x/(1 - d(0, n)*x/(1 - d(1, n)*x/(1 - d(2, n)*x/(...))))).

%F d(m, n)*d(m+1, n) is a rational solution in x of the elliptic equation y^2 = -4*x^3 + ((n+1)^2 + 8)*x^2 - 2*(n+3)*x + 1. The division polynomials for multiples of the point with x = 1, correspondent to the Hankel transform of row n in the array T(n, k).

%F T(n, k + 2) = Sum_{j >= 0} A377443(k, j)*n^j. This polynomial starts with A000108(k+2) + A371965(k+2)*n + ..., where A371965 is known to count peaks in the set of Catalan words of length k.

%e The array begins:

%e [0] 1, 1, 2, 5, 14, 42, 132, 429, 1430, ... = A000108

%e [1] 1, 1, 2, 6, 21, 78, 299, 1172, 4677, ... = A254316

%e [2] 1, 1, 2, 7, 30, 136, 630, 2959, 14058, ...

%e [3] 1, 1, 2, 8, 41, 222, 1221, 6774, 37853, ...

%e [4] 1, 1, 2, 9, 54, 342, 2192, 14129, 91494, ...

%e [5] 1, 1, 2, 10, 69, 502, 3687, 27184, 201045, ...

%o (PARI)

%o T(n, max_k) = Vec(-2*((n+1)*x-1)/((x-1)*(n*x-1)+((n*x^2-(n+1)*x+1)^2-4*x*(x-1)*((n+1)*x-1)+O(x^max_k))^(1/2)))

%Y Cf. A377442 (extension for -n), A105633 (row -1), A152172 (row -2).

%Y Cf. A000108 (row 0), A254316 (row 1).

%Y Cf. A000012 (Hankel transform of row 0), A006720 (Hankel transform of row 1).

%Y Cf. A330025 (Hankel transform of row -1), A328380 (Hankel transform of row -2).

%Y Cf. A371965, A377443.

%K nonn,tabl

%O 0,6

%A _Thomas Scheuerle_, Oct 28 2024