A212425 - OEIS

%I #18 Nov 16 2024 20:19:03

%S 1,3,30,406,6336,107415,1922310,35739990,683593902,13364444808,

%T 265869803598,5364752267064,109533577804350,2258715717810522,

%U 46974966620274810,984153696477302700,20751365954898103338,440033530633057730880,9377869165352931696930

%N G.f. satisfies: A(x) = ( x + A(A(x)) )^3 where g.f. A(x) = Sum_{n>=1} a(n)*x^(8*n-5).

%C Conjecture: (2*n-1) divides a(n); see A212426.

%C More generally, we have the conjecture:

%C If    A(x) = ( x + A(A(x)) )^b

%C where A(x) = Sum_{n>=1} a(n) * x^((b^2-1)*(n-1)+b)

%C then  ((b-1)*(n-1)+1) divides a(n).

%H Alois P. Heinz, <a href="/A212425/b212425.txt">Table of n, a(n) for n = 1..100</a>

%F G.f.: A(x) = d/dx G(x^8)/(4*x^4) where G(x) = Sum_{n>=1} A212426(n)*x^n is the g.f. of A212426.

%F a(n) = (2*n-1)*A212426(n).

%F 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]

%e G.f.: A(x) = x^3 + 3*x^11 + 30*x^19 + 406*x^27 + 6336*x^35 + 107415*x^43 +...

%e such that A(x) = (x + A(A(x)))^3, where

%e 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 +...

%e Note that A(A(x))^(1/3) = A(x) + A(A(A(x))), where

%e 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 +...

%e 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 +...

%p A:= proc(n) option remember;

%p       `if`(n=1, unapply(x, x), unapply (convert (series

%p        ((x+(A(n-1)@@2)(x))^3, x, n+10), polynom), x))

%p     end:

%p a:= n-> coeff (A(8*n-5)(x), x, 8*n-5):

%p seq (a(n), n=1..30);  # _Alois P. Heinz_, May 17 2012

%t 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_ *)

%o (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)}

%o for(n=1, 30, print1(a(n), ", "))

%o (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);

%o makelist(T(n,1),n,1,20); /* _Vladimir Kruchinin_, May 17 2012 */

%Y Cf. A212426, A212392.

%K nonn

%O 1,2

%A _Paul D. Hanna_, May 16 2012