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 A124753 a(3n+k) = (k+1)*binomial(4n+k, n)/(3n+k+1), where k is n reduced mod 3. 5
 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 (list; graph; refs; listen; history; text; internal format)
 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 \./ .|....\|/ . [David Callan, Aug 22 2014] LINKS J.-B. Priez, 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). 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 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 CROSSREFS Cf. A084080, A118968, A002293, A069271 (trisection), A006632 (trisection. 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

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Last modified September 26 05:07 EDT 2021. Contains 347664 sequences. (Running on oeis4.)