A124753 a(3n+k) = (k+1)*binomial(4n+k, n)/(3n+k+1), where k is n reduced mod 3.

1, 1, 1, 1, 2, 3, 4, 9, 15, 22, 52, 91, 140, 340, 612, 969, 2394, 4389, 7084, 17710, 32890, 53820, 135720, 254475, 420732, 1068012, 2017356, 3362260, 8579560, 16301164, 27343888, 70068713, 133767543, 225568798, 580034052, 1111731933, 1882933364, 4855986044, 9338434700

Offset

0,5

Comments

Row sums of Riordan array (1,x(1-x^3))^(-1). Also row sums of A124752.

a(n) is the number of ordered trees (A000108) with n vertices in which every non-leaf non-root vertex has exactly two children that are leaves. For example, a(4) counts the 2 trees

\ /

| and \|/ . - David Callan, Aug 22 2014

Coefficients of the generating function E(1,t*E(4,t^3)) that appears on line 4 of Table 2 on page 210 of the Hering link. E(d,t)=1+t*E(d,t)^d is the generating function for Fuss-Catalan numbers. - Robert A. Russell, May 21 2026

Links

Table of n, a(n) for n=0..38.

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

J.-B. Priez and A. Virmaux, Non-commutative Frobenius characteristic of generalized parking functions: Application to enumeration, arXiv:1411.4161 [math.CO], 2014-2015.

Formula

a(3n) = A002293(n), a(3n+1) = A069271(n), a(3n+2) = A006632(n+1).

a(n) = ((mod(n,3)+1)*C(4*floor(n/3)+mod(n,3), floor(n/3))/ (3*floor(n/3) + 1 + mod(n, 3))). - Paul Barry, Dec 14 2006

G.f. satisfies: A(x) = 1 + x*A(x)^2*A(w*x)*A(w^2*x), where w = exp(2*Pi*I/3). - Paul D. Hanna, Jun 04 2012

G.f. satisfies: A(x) = 1 + x*A(x)*G(x^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293. - Paul D. Hanna, Jun 04 2012

Conjecture: +8019*n*(n-1)*(n+1)*a(n) +17496*n*(n-1)*(n-3)*a(n-1) +2592*(3*n-5)*(n-1)*(3*n-16)*a(n-2) +216*(-224*n^3+48*n^2+3926*n-6331)*a(n-3) +576*(-288*n^3+2448*n^2-6558*n+5443)*a(n-4) +768*(-288*n^3+3600*n^2-14878*n+20375)*a(n-5) -8192*(4*n-23)*(2*n-11)*(4*n-21)*a(n-6)=0. - R. J. Mathar, Oct 30 2014

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/3)} a(3*k) * a(n-1-3*k). - Seiichi Manyama, Jul 07 2025

Maple

A124753 := proc(n)
    local k, np;
    k := modp(n, 3) ;
    np := floor(n/3) ;
    (k+1)*binomial(np+n, np)/(n+1) ;
end proc:
seq(A124753(n), n=0..40) ; # R. J. Mathar, Oct 30 2014

Mathematica

a[n_] := Module[{q, k}, {q, k} = QuotientRemainder[n, 3]; (k+1)*Binomial[4q + k, q]/(3q + k + 1)];
Table[a[n], {n, 0, 34}] (* Jean-François Alcover, Nov 20 2017 *)

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*A*exp(sum(m=1, n\3, 3*polcoeff(log(A+x*O(x^n)), 3*m)*x^(3*m))+x*O(x^n))); polcoeff(A, n)} \\ Paul D. Hanna, Jun 04 2012
(PARI) apr(n, p, r) = r*binomial(n*p+r, n)/(n*p+r);
a(n) = apr(n\3, 4, n%3+1); \\ Seiichi Manyama, Jul 20 2025

Crossrefs

Cf. A084080, A002293, A069271 (trisection), A006632 (trisection).

Cf. A047749, A118968.

Sequence in context: A033076 A121431 A084080 * A248647 A284437 A049909

Adjacent sequences: A124750 A124751 A124752 * A124754 A124755 A124756

Keyword

nonn,easy

Author

Paul Barry, Nov 06 2006

Status

approved