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A272484
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G.f. A(x) satisfies: A( A(x)^3 ) = C(x) * A(x)^2, where C(x) = x + C(x)^2, with A(0)=0, A'(0)=1.
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2
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1, 1, 2, 4, 10, 28, 86, 278, 928, 3164, 10958, 38428, 136168, 486796, 1753660, 6359961, 23202408, 85093552, 313548346, 1160248084, 4309812532, 16064728072, 60070599076, 225271863550, 847042748378, 3192758928650, 12061704111576, 45662648135238, 173204482763760, 658180582310888, 2505341336035650, 9551632787000829, 36469897605758744, 139443687986144472, 533869533407865024, 2046496258409861740, 7854102611559917914
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OFFSET
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1,3
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COMMENTS
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The radius of convergence of g.f. A(x) is 1/4.
Specific value S = A(1/4) = 0.44982760488955294204795759797171897522321034552221... satisfies:
(1) S^2 = 2 * A(S^3),
(2) S^4 = 8 * A(S^6/8) / (1 - sqrt(1 - 4*S^3)).
Limit a(n)/A000108(n-1) appears to be near 0.6564...
The numerical value of this limit is 0.6564415409950121... . - Vaclav Kotesovec, May 07 2016
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LINKS
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FORMULA
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G.f. A(x) satisfies:
(1) A( A(x-x^2)^3 ) = x * A(x-x^2)^2.
(2) A(x-x^2) = Series_Reversion( A(x^3)/x^2 ).
(3) A( A(x^3)/x^2 - A(x^3)^2/x^4 ) = x.
a(n) ~ c * 4^n / n^(3/2), where c = 0.09258936990935582... . - Vaclav Kotesovec, May 07 2016
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EXAMPLE
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G.f.: A(x) = x + x^2 + 2*x^3 + 4*x^4 + 10*x^5 + 28*x^6 + 86*x^7 + 278*x^8 + 928*x^9 + 3164*x^10 + 10958*x^11 + 38428*x^12 +...
such that A( A(x)^3 ) = C(x) * A(x)^2, where C(x) = x + C(x)^2.
RELATED SERIES.
C(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + 429*x^8 + 1430*x^9 + 4862*x^10 + 16796*x^11 + 58786*x^12 +...+ A000108(n-1)*x^n +...
A(x)^2 = x^2 + 2*x^3 + 5*x^4 + 12*x^5 + 32*x^6 + 92*x^7 + 284*x^8 + 920*x^9 + 3080*x^10 + 10544*x^11 + 36684*x^12 + 129228*x^13 + 459860*x^14 +...
A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 25*x^6 + 72*x^7 + 216*x^8 + 680*x^9 + 2226*x^10 + 7506*x^11 + 25858*x^12 + 90498*x^13 + 320580*x^14 + 1146670*x^15 +...
A( A(x)^3 ) = x^3 + 3*x^4 + 9*x^5 + 26*x^6 + 78*x^7 + 243*x^8 + 786*x^9 + 2619*x^10 + 8928*x^11 + 30967*x^12 + 108870*x^13 + 386928*x^14 + 1387560*x^15 +...
where A( A(x)^3 ) = C(x)*A(x)^2.
A(x-x^2) = x - x^4 + 2*x^7 - 4*x^10 + 4*x^13 + 23*x^16 - 212*x^19 + 1148*x^22 - 4906*x^25 + 16904*x^28 - 41046*x^31 + 6730*x^34 + 713246*x^37 - 5703472*x^40 +...
where A(x-x^2) = Series_Reversion( A(x^3)/x^2 ).
A( A(x-x^2)^3 ) = x^3 - 2*x^6 + 5*x^9 - 12*x^12 + 20*x^15 + 22*x^18 - 438*x^21 + 2780*x^24 - 13124*x^27 + 50092*x^30 - 145875*x^33 + 201848*x^36 +...
where A( A(x-x^2)^3 ) = x * A(x-x^2)^2.
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PROG
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(PARI) {a(n) = my(A=x, C=x, X=x+x*O(x^n)); for(i=1, n, C = X + C^2; A = (2*A - subst(A, x, A^3)/(C*A) )); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))
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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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