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A212392 G.f. satisfies: A(x) = (x + A(A(x)))^2 where g.f. A(x) = Sum_{n>=1} a(n)*x^(3*n-1). 4

%I

%S 1,2,9,56,400,3096,25256,213832,1861272,16552320,149737632,1373597892,

%T 12747475260,119465392536,1129016386080,10747541655584,

%U 102960795706704,991886971036248,9603034303017640,93386133268757760,911779906476551616,8934398271363272642

%N G.f. satisfies: A(x) = (x + A(A(x)))^2 where g.f. A(x) = Sum_{n>=1} a(n)*x^(3*n-1).

%C Conjecture: n divides a(n); see A212391.

%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 Paul D. Hanna, <a href="/A212392/b212392.txt">Table of n, a(n) for n = 1..200</a>

%F a(n) = n*A212391(n).

%e G.f.: A(x) = x^2 + 2*x^5 + 9*x^8 + 56*x^11 + 400*x^14 + 3096*x^17 + 25256*x^20 +...

%e such that A(x) = (x + A(A(x)))^2, where

%e A(A(x)) = x^4 + 4*x^7 + 24*x^10 + 168*x^13 + 1284*x^16 + 10384*x^19 + 87364*x^22 + 756808*x^25 + 6704968*x^28 + 60471040*x^31 +...+ A212277(n+1)*x^(3*n+1) +...

%e Note that sqrt(A(A(x))) = A(x) + A(A(A(x))), where

%e sqrt(A(A(x))) = x^2 + 2*x^5 + 10*x^8 + 64*x^11 + 464*x^14 + 3624*x^17 + 29746*x^20 + 252976*x^23 + 2209488*x^26 + 19701504*x^29 +...

%e A(A(A(x))) = x^8 + 8*x^11 + 64*x^14 + 528*x^17 + 4490*x^20 + 39144*x^23 + 348216*x^26 + 3149184*x^29 + 28872401*x^32 +...

%o (PARI) {a(n)=local(A=x^2+x^3);for(i=1,n,A=(x+subst(A,x,A+O(x^(3*n))))^2);polcoeff(A,3*n-1)}

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

%o (Maxima) A(n,k):= if n<2*k then 0 else if n/2=k then 1 else sum(binomial(2*k,j)*sum(A(i,2*k-j)*A(n-j,i),i,2*k-j+1,n-j-1),j,0,2*k-1);

%o makelist(A(n,1),n,1,17); [_Vladimir Kruchinin_, May 15 2012]

%Y Cf. A212391, A212277.

%K nonn

%O 1,2

%A _Paul D. Hanna_, May 12 2012

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Last modified May 24 04:25 EDT 2019. Contains 323528 sequences. (Running on oeis4.)