A391234 Diagonal of the 3-dimensional geode array: a(n) = G(n,n,n).

1, 319, 669123, 2297259900, 9943305738520, 49263623024877534, 266686057181713977780, 1536972107960438305997148, 9280523146533806331987399720, 58094384066172033744456805634550, 374253836760702740599317449688465810, 2468122083457731896636582178758135814672

Offset

0,2

Comments

The Geode G is a 3-dimensional array of positive integers introduced by Wildberger and Rubine (2025) via operator inversion of the hyper-Catalan generating series. For the 3-dimensional geode, S - 1 = (t_2 + t_3 + t_4) * G where S is the hyper-Catalan series in variables t_2, t_3, t_4. This sequence records the main diagonal a(n) = G(n,n,n).

Every entry G(m2,m3,m4) can be expressed as a closed-form alternating binomial sum of hyper-Catalan numbers (Kotlarz, 2026): G(m2,m3,m4) = Sum_{i=0..m3} Sum_{j=0..m4} (-1)^{i+j} * C(i+j,i) * C(m2+1+i+j, m3-i, m4-j), where C(m2,m3,m4) = (S+L-1)!/(m2!*m3!*m4!*L!) with S = m2+m3+m4, L = m2+2*m3+3*m4+1.

Combinatorial interpretation: G(m2,m3,m4) counts exactly the ballot sequences (Lukasiewicz words) of type (m2+1,m3,m4) in which the first negative step is -1. Equivalently, no step of value -k (k >= 2) precedes the first step of value -1. Equivalently, in the corresponding ordered tree, no node of arity r >= 3 whose first r-2 children are all leaves precedes the first binary node in pre-order.

Amdeberhan, Kauers, and Zeilberger (arXiv:2508.10245) found a second-order holonomic recurrence for this sequence with degree-35 polynomial coefficients.

Links

Table of n, a(n) for n=0..11.

T. Amdeberhan, M. Kauers, and D. Zeilberger, The Challenge of Computing Geode Numbers, arXiv:2508.10245 [math.CO], 2025; Palestine J. Math. 14(4) (2025), 709-713.

T. Amdeberhan and D. Zeilberger, Proofs Of Three Geode Conjectures, arXiv:2506.17862 [math.CO], 2025.

I. M. Gessel, Lattice Paths and the Geode, arXiv:2507.09405 [math.CO], 2025.

M. Gossow, Ordered Trees and the Geode, arXiv:2507.18097 [math.CO], 2025.

D. Rubine, Computing the 4D Geode, arXiv:2512.21785 [math.CO], 2025.

D. Rubine, Hyper-Catalan and Geode Recurrences and Three Conjectures of Wildberger, arXiv:2507.04552 [math.CO], 2025.

N. J. Wildberger and D. Rubine, A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, Amer. Math. Monthly, 132(5) (2025), 420-437.

Formula

a(n) = G(n,n,n) where G(m2,m3,m4) = Sum_{i=0..m3} Sum_{j=0..m4} (-1)^{i+j} * binomial(i+j,i) * C(m2+1+i+j, m3-i, m4-j), where C(m2,m3,m4) = (S+L-1)!/(m2!*m3!*m4!*L!) with S = m2+m3+m4, L = m2+2*m3+3*m4+1.

Equivalently, G satisfies the recurrence G(m2,m3,m4) = C(m2+1,m3,m4) - G(m2+1,m3-1,m4) - G(m2+1,m3,m4-1) (indices below 0 give 0).

Prog

(Python)
from math import factorial
def C(m2, m3, m4):
    S, L = m2+m3+m4, m2+2*m3+3*m4+1
    return factorial(S+L-1) // (factorial(m2)*factorial(m3)*factorial(m4)*factorial(L))
def a(n):
    res = 0
    for i in range(n+1):
        for j in range(n+1):
            t = factorial(i+j)//(factorial(i)*factorial(j)) * C(n+1+i+j, n-i, n-j)
            res += t if (i+j)%2==0 else -t
    return res
print([a(n) for n in range(12)])
(PARI) C(m2, m3, m4) = my(S=m2+m3+m4, L=m2+2*m3+3*m4+1); (S+L-1)!/(m2!*m3!*m4!*L!);
a(n) = sum(i=0, n, sum(j=0, n, (-1)^(i+j) * binomial(i+j, i) * C(n+1+i+j, n-i, n-j))); \\ Michel Marcus, Mar 16 2026

Crossrefs

Cf. A000108 (Catalan numbers, first column of 2-dimensional Geode), A001764 (Fuss-Catalan), A383439 (diagonal of 2-dimensional Geode), A104978 (hyper-Catalan triangle).

Sequence in context: A064905 A293921 A121010 * A110289 A258681 A258674

Adjacent sequences: A391231 A391232 A391233 * A391235 A391236 A391237

Keyword

nonn,easy

Author

Steven Kotlarz, Mar 15 2026

Status

approved