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

Offset: 0

Author

Thomas Scheuerle, Oct 28 2024

Keywords

• nonn
• tabl

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.

Examples

			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, ...
		

Crossrefs

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

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

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

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

Cf. A371965, A377443.

Programs

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)))

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.