A129875 Sequence t_n arising in enumeration of arrays of directed blocks (see Quaintance reference for precise definition).

0, 2, 4, 9, 22, 45, 120, 239, 670, 1320, 3824, 7494, 22224, 43428, 131076, 255711, 782494, 1525014, 4718676, 9190677, 28698678, 55876161, 175821684, 342247020, 1083990696, 2109783728, 6720103072, 13078616688, 41863938704, 81473918808, 261928330164, 509761089231, 1645149181278

Offset

1,2

Links

Table of n, a(n) for n=1..33.

Jocelyn Quaintance, Letter Representations of m x n x p Proper Arrays, arXiv:math/0412244 [math.CO], 2004-2006. See Table 1.

Jocelyn Quaintance, Combinatoric Enumeration of Two-Dimensional Proper Arrays, Discrete Math., 307 (2007), 1844-1864. See Table 1.

Formula

See Quaintance reference for generating functions that produce A129872-A129886.

From Thomas Scheuerle, Jan 14 2026: (Start)

The generating function A(x) satisfies: (8*x^2 - x^2)*A(x)^3 + (12*x^4 + 26*x^3 - 3*x^2 - 4*x)*A(x)^2 + (10*x^4 + 16*x^3 + 19*x^2 - 2*x - 3)*A(x) + (-8*x^5 - 3*x^4 + 16*x^3 + 6*x^2) = 0.

a(n) = Sum_{k=1..floor(n/2)} A129873(n - 2*k)*(2/k)*binomial(3*k-3, k-1), with A129873(0) = 1. (End)

Prog

(PARI)
listA129873(m) = {my(S = t*O(t)); for (n= 1, m+1, S = (- 8*t - 3 - (8*t^4 - t^2)*S^3 - (24*t^3 - 2*t^2 - 4*t)*S^2)/(24*t^2 + 2*t - 3); ); return(vector(m, i , polcoeff(S, i, t))); }
a(n) = { my(a129873 = concat([1], listA129873(n*2))); sum(k=1, floor(n/2), a129873[1+n-2*k]*(2/k)*binomial(3*k-3, k-1)); } \\ Thomas Scheuerle, Jan 14 2026

Crossrefs

Cf. A129873-A129886.

Sequence in context: A156801 A363771 A057580 * A055094 A369472 A055729

Adjacent sequences: A129872 A129873 A129874 * A129876 A129877 A129878

Keyword

nonn

Author

N. J. A. Sloane, May 26 2007

Extensions

More terms from Thomas Scheuerle, Jan 14 2026

Status

approved