OFFSET
1,2
COMMENTS
Conjecture: (2*n-1) divides a(n); see A212426.
More generally, we have the conjecture:
If A(x) = ( x + A(A(x)) )^b
where A(x) = Sum_{n>=1} a(n) * x^((b^2-1)*(n-1)+b)
then ((b-1)*(n-1)+1) divides a(n).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..100
FORMULA
a(n) = (2*n-1)*A212426(n).
a(n) = T(8*n-5,1), T(n,k) = if n<3*k then 0 else if n/3=k then 1 else sum(j=0..3*k-1, C(3*k,j)*sum(i=3*k-j+1..n-j-1, T(i,3*k-j)*T(n-j,i))). [Vladimir Kruchinin, May 17 2012]
EXAMPLE
G.f.: A(x) = x^3 + 3*x^11 + 30*x^19 + 406*x^27 + 6336*x^35 + 107415*x^43 +...
such that A(x) = (x + A(A(x)))^3, where
A(A(x)) = x^9 + 9*x^17 + 117*x^25 + 1788*x^33 + 29925*x^41 + 530910*x^49 + 9809193*x^57 + 186734493*x^65 + 3637247445*x^73 +...
Note that A(A(x))^(1/3) = A(x) + A(A(A(x))), where
A(A(x))^(1/3) = x^3 + 3*x^11 + 30*x^19 + 407*x^27 + 6363*x^35 + 108009*x^43 + 1934721*x^51 + 35995815*x^59 + 688861845*x^67 +...
A(A(A(x))) = x^27 + 27*x^35 + 594*x^43 + 12411*x^51 + 255825*x^59 + 5267943*x^67 + 108864873*x^75 + 2261456685*x^83 +...
MAPLE
A:= proc(n) option remember;
`if`(n=1, unapply(x, x), unapply (convert (series
((x+(A(n-1)@@2)(x))^3, x, n+10), polynom), x))
end:
a:= n-> coeff (A(8*n-5)(x), x, 8*n-5):
seq (a(n), n=1..30); # Alois P. Heinz, May 17 2012
MATHEMATICA
T[n_, k_] := T[n, k] = If[n<3*k, 0, If[n/3 == k, 1, Sum[Binomial[3*k, j]*Sum[T[i, 3*k-j]*T[n-j, i], {i, 3*k-j+1, n-j-1}], {j, 0, 3*k-1}]]]; Table[T[8*n-5, 1], {n, 1, 19 }] (* Jean-François Alcover, Feb 14 2014, after Vladimir Kruchinin *)
PROG
(PARI) {a(n)=local(A=x^3+3*x^11); for(i=1, n, A=(x+subst(A, x, A+O(x^(8*n))))^3); polcoeff(A, 8*n-5)}
for(n=1, 30, print1(a(n), ", "))
(Maxima) T(n, k):= if n<3*k then 0 else if n/3=k then 1 else sum(binomial(3*k, j)*sum(T(i, 3*k-j)*T(n-j, i), i, 3*k-j+1, n-j-1), j, 0, 3*k-1);
makelist(T(n, 1), n, 1, 20); /* Vladimir Kruchinin, May 17 2012 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 16 2012
STATUS
approved