A393769 Coefficients of E(2,t*E(4,t^2)), where E(d,t)=1+t*E(d,t)^d.

1, 1, 2, 6, 18, 61, 206, 736, 2634, 9720, 36024, 135994, 516030, 1981642, 7648260, 29764416, 116364138, 457689632, 1807445576, 7170850408, 28548301000, 114073426233, 457173621678, 1837717521408, 7406075729478, 29921472956312, 121154164282920, 491604207872268

Offset

0,3

Comments

These are the coefficients of the generating function E(2,t*E(4,t^2)) that appears on line 2 of Table 2 on page 210 of the Hering link. It is used in enumerating achiral polyominoes of the hyperbolic regular tiling with Schläfli symbol {3,3,3,oo}. Note that E(d,t) is the generating function for Fuss-Catalan numbers.

Links

Table of n, a(n) for n=0..27.

F. Hering, R. C. Read, and G. C. Shephard, The enumeration of stack polytopes and simplicial clusters, Discrete Math., 40 (1982), 203-217.

Formula

a(n) ~ 3^(1/4) * 2^(5*n-1) / (sqrt(Pi) * n^(3/2) * (1 + sqrt(3))^(2*n - 1/2)). - Vaclav Kotesovec, May 22 2026

Mathematica

Prepend[Table[Sum[Binomial[2n-4m, n-2m+1]Binomial[n+2m-1, m]/(n+m), {m, 0, (n-1)/2}], {n, 1, 30}], 1]

Crossrefs

Cf. A182299.

Sequence in context: A357693 A150047 A386254 * A150048 A363511 A150049

Adjacent sequences: A393766 A393767 A393768 * A393770 A393771 A393772

Keyword

nonn

Author

Robert A. Russell, May 21 2026

Status

approved