A129874 Sequence h_n arising in enumeration of arrays of directed blocks (see Quaintance reference for precise definition).

2, 1, 4, 4, 16, 19, 76, 98, 392, 531, 2124, 2974, 11896, 17060, 68240, 99658, 398632, 590563, 2362252, 3540464, 14161856, 21430267, 85721068, 130771376, 523085504, 803538100, 3214152400, 4967127736, 19868510944, 30866224824, 123464899296

Offset

1,1

Links

Table of n, a(n) for n=1..31.

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 13 2026: (Start)

The generating function A(x) satisfies: (8*x^2 - 1)*A(x)^3 + (12*x^2 - 2*x - 2)*A(x)^2 + (10*x^2 - 1)*A(x) + (8*x^3 + 9*x^2 + 2*x) = 0.

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

a(2*n+1) = (Sum_{k=1..n} A129873(2*n - 2*k+1)*(binomial(3*k-3, k-1)/(2*k-1))) + 2*Sum_{r=1..n} (binomial(3*(n-r)+1, n-r)/((n-r)+1))*(binomial(3*r-3, r-1)/(2*r-1)), for n > 0 and 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))); }
h_even(n, a129873) = {sum(k=1, n, a129873[1+2*n-2*k]*(binomial(3*k-3, k-1)/(2*k-1))); }
h_odd(n, a129873) = {sum(k=1, n, a129873[1+2*n-2*k+1]*(binomial(3*k-3, k-1)/(2*k-1)))+2*sum(r=1, n, (binomial(3*(n-r)+1, n-r)/((n-r)+1))*(binomial(3*r-3, r-1)/(2*r-1))); }
a(n) = {if(n==1, 2, my(a129873 = concat([1], listA129873(n))); if(n%2==0, h_even(n/2, a129873), h_odd((n-1)/2, a129873))); }

Crossrefs

Cf. A129873-A129886.

Sequence in context: A057772 A145861 A191660 * A021983 A349994 A375123

Adjacent sequences: A129871 A129872 A129873 * A129875 A129876 A129877

Keyword

nonn

Author

N. J. A. Sloane, May 26 2007

Extensions

More terms from Thomas Scheuerle, Jan 13 2026

Status

approved