A377441 - OEIS

A377441

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)).

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

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OFFSET

0,6

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..85.

FORMULA

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).

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/(...))))).

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).

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.

EXAMPLE

The array begins:

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

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

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

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

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

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

PROG

(PARI)

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)))

CROSSREFS