OFFSET
0,3
COMMENTS
Compare to: G(x)^3 = 1 + x*G(x) + x*G(x)^2 + x*G(x)^3 holds when G(x) = 1/(1-x).
From Yu Hin Au, Jan 19 2026: (Start)
a(n) = number of operator monomials M in a 4-ary associative algebra with a unary linear operator L, where M has n-1 total operations. E.g., the a(3) = 7 such operator monomials are L(L(a)), L(abcd), L(a)bcd, aL(b)cd, abL(c)d, abcL(d), and abcdefg.
a(n) = number of labeled Schröder paths of semilength n-1 avoiding DDDD where each descent (maximal sequence run of down steps) of length l>=2 is labeled by a (l-1)-subset of {1,2}. E.g., the a(3) = 7 such paths are UU(DD)_{1}, UU(DD)_{2}, 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 3 nonnegative parts (the final descent is unlabeled). E.g., the a(3) = 7 such paths are
- UUUDDD, UUDUDD, UDUUDD, UDUDUD;
- UU(DD)_SUD where S belongs to {(1,0,0), (0,1,0), (0,0,1)}. (End)
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
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) ~ sqrt(1 + sqrt((9 + 40*sqrt(3))/13)) * (9 + 6*sqrt(3) + sqrt(153 + 100*sqrt(3)))^n / (sqrt(Pi) * n^(3/2) * 2^(n + 3/2) * 3^(n + 3/4)). - Vaclav Kotesovec, Aug 11 2021
a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(n+2*k,n-1-k) for n > 0. - Seiichi Manyama, Aug 05 2023
G.f.: 1 + Series_Reversion( x / (1 + x + x * (1 + x)^3) ). - Seiichi Manyama, Oct 03 2025
G.f.: 1/(1 - x*B(x)), where B(x) is the g.f. of A161797. - Seiichi Manyama, Dec 15 2025
G.f.: A(x) satisfies A(x) = 1 + x*A(x)*(1 - A(x)^2 + A(x)^3). - Yu Hin Au, Jan 19 2026
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 29*x^4 + 131*x^5 + 627*x^6 + 3124*x^7 + 16032*x^8 + 84162*x^9 + 449828*x^10 + 2439550*x^11 + 13391105*x^12 + ...
RELATED SERIES.
A(x)^2 = 1 + 2*x + 5*x^2 + 18*x^3 + 76*x^4 + 348*x^5 + 1681*x^6 + 8432*x^7 + 43495*x^8 + 229260*x^9 + 1229371*x^10 + ...
A(x)^6 = 1 + 6*x + 27*x^2 + 122*x^3 + 579*x^4 + 2862*x^5 + 14588*x^6 + 76146*x^7 + 405039*x^8 + 2187756*x^9 + 11967426*x^10 + ...
A(x)^3 = 1 + 3*x + 9*x^2 + 34*x^3 + 147*x^4 + 684*x^5 + 3341*x^6 + 16896*x^7 + 87702*x^8 + 464566*x^9 + 2501178*x^10 + ...
MATHEMATICA
{1}~Join~Table[(1/n)*Sum[Binomial[n, k]*Binomial[n + 2*k, n - 1 - k], {k, 0, n - 1}], {n, 28}] (* Michael De Vlieger, Nov 26 2025 *)
PROG
(PARI) {a(n) = my(A=1); for(i=1, n, A = (1 + x*A + x*A^2 + x*A^6 +x*O(x^n))^(1/3) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 25 2018
STATUS
approved