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A352384
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G.f. A(x) satisfies: A(x) = (1 + 2*x*A(x))^6 / (1 + x*A(x))^3.
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2
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1, 9, 111, 1581, 24468, 399735, 6784186, 118444293, 2113587804, 38377421060, 706774205943, 13170180868299, 247862354439196, 4704490506021162, 89949748461476772, 1730889637195688117, 33495746280466024908
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OFFSET
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0,2
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COMMENTS
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Self-convolution cube root yields A352383.
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LINKS
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FORMULA
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G.f. A(x) satisfies:
(1) A(x) = (1 + 2*x*A(x))^6 / (1 + x*A(x))^3.
(2) A( x*(1+x)^3/(1+2*x)^6 ) = (1+2*x)^6/(1+x)^3.
(3) A(x) = (1/x) * Series_Reversion( x*(1+x)^3/(1+2*x)^6 ).
(4) x = ( A(x)^(1/3) - 4 + sqrt( A(x)^(2/3) + 8*A(x)^(1/3) ) ) / (8*A(x)).
(5) 0 = 4*A(x)^2*x^2 + (4 - A(x)^(1/3))*A(x)*x + (1 - A(x)^(1/3)).
a(n) ~ sqrt(278513 - 1003421/sqrt(13)) * 2^(3*n + 3/2) * (587 - 143*sqrt(13))^n / (sqrt(Pi) * n^(3/2) * 3^(3*n + 5/2)). - Vaclav Kotesovec, Mar 15 2022
D-finite with recurrence 3*n *(3*n+2) *(2*n+3) *(3*n+1) *(5114882323*n -3577270936)*(n+1)*a(n) -8*n*(12009974028164*n^5 +15575274162434*n^4 +20452834455*n^3 -7391009529770*n^2 -2779978786544*n -151626455514) *a(n-1) +64*(29840008960856*n^6 -7909817331616*n^5 -21378617546230*n^4 -22395081360175*n^3 +39992783684339*n^2 -16585158398179*n +2497181632755) *a(n-2) -4608*(6*n-11) *(6*n-7) *(3*n-4) *(2*n-3) *(3*n-5) *(4265440*n -810084569)*a(n-3)=0. - R. J. Mathar, Jul 20 2023
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EXAMPLE
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G.f.: A(x) = 1 + 9*x + 111*x^2 + 1581*x^3 + 24468*x^4 + 399735*x^5 + 6784186*x^6 + 118444293*x^7 + 2113587804*x^8 + 38377421060*x^9 + ...
where
A(x)^(1/3) = (1 + 2*x*A(x))^2/(1 + x*A(x)) = 1 + 3*x + 28*x^2 + 350*x^3 + 5020*x^4 + 78023*x^5 + 1278340*x^6 + ... + A352383(n)*x^n + ...
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MATHEMATICA
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CoefficientList[(InverseSeries[Series[x*(1 + x)^3/(1 + 2*x)^6, {x, 0, 20}], x]/x), x] (* Vaclav Kotesovec, Mar 15 2022 *)
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PROG
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(PARI) /* Using Series Reversion */
{a(n) = my(A = (1/x)*serreverse( x*(1+x)^3/(1+2*x +x^2*O(x^n))^6 )); polcoeff(A, n)}
for(n=0, 30, 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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