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A212392
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G.f. satisfies: A(x) = (x + A(A(x)))^2 where g.f. A(x) = Sum_{n>=1} a(n)*x^(3*n-1).
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4
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1, 2, 9, 56, 400, 3096, 25256, 213832, 1861272, 16552320, 149737632, 1373597892, 12747475260, 119465392536, 1129016386080, 10747541655584, 102960795706704, 991886971036248, 9603034303017640, 93386133268757760, 911779906476551616, 8934398271363272642
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OFFSET
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1,2
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COMMENTS
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Conjecture: n divides a(n); see A212391.
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).
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LINKS
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FORMULA
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EXAMPLE
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G.f.: A(x) = x^2 + 2*x^5 + 9*x^8 + 56*x^11 + 400*x^14 + 3096*x^17 + 25256*x^20 +...
such that A(x) = (x + A(A(x)))^2, where
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) +...
Note that sqrt(A(A(x))) = A(x) + A(A(A(x))), where
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 +...
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 +...
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PROG
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(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)}
for(n=1, 30, print1(a(n), ", "))
(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);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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