OFFSET
0,3
COMMENTS
From Yu Hin Au, Jan 19 2026: (Start)
a(n) = number of operator monomials M in a 5-ary associative algebra with a unary linear operator L, where M has n-1 total operations. E.g., the a(3) = 8 such operator monomials are L(L(a)), L(abcde), L(a)bcde, aL(b)cde, abL(c)de, abcL(d)e, abcdL(e), and abcdefghi.
a(n) = number of labeled Schröder paths of semilength n-1 avoiding DDDDD where each descent (maximal sequence run of down steps) of length l >=2 is labeled by a (l-1)-subset of {1,2,3}. E.g., the a(3) = 8 such paths are UU(DD)_{1}, UU(DD)_{2}, UU(DD)_{3}, UHD, UDUD, UDH, HUD, and HH.
a(n) = number of labeled Dyck paths of semilength n, where every non-terminal descent of length l>=2 is labeled by a composition of l-1 consisting of 4 nonnegative parts (the final descent is unlabeled). E.g., the a(3) = 8 such paths are
- UUUDDD, UUDUDD, UDUUDD, UDUDUD;
- UU(DD)_SUD where S belongs to {(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1)}. (End)
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..1000
Yu Hin Au and Murray R. Bremner, A new generalization of the Narayana numbers inspired by linear operators on associative d-ary algebras, arXiv:2511.13671 [math.CO], 2025. See p. 2, Table 2.
FORMULA
a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(n+3*k,n-1-k) for n > 0.
G.f.: 1 + Series_Reversion( x / (1 + x + x * (1 + x)^4) ). - Seiichi Manyama, Oct 03 2025
G.f.: 1/(1 - x*B(x)), where B(x) is the g.f. of A321798. - Seiichi Manyama, Dec 15 2025
MATHEMATICA
{1}~Join~Table[(1/n)*Sum[Binomial[n, k]*Binomial[n + 3*k, n - 1 - k], {k, 0, n - 1}], {n, 23}] (* Michael De Vlieger, Nov 26 2025 *)
PROG
(PARI) a(n) = if(n==0, 1, sum(k=0, n-1, binomial(n, k)*binomial(n+3*k, n-1-k))/n);
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Aug 05 2023
STATUS
approved