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A300989
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O.g.f. A(x) satisfies: A(x) = x*(1 - 3*x*A'(x)) / (1 - 4*x*A'(x)).
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7
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1, 1, 6, 50, 520, 6312, 86080, 1288704, 20862720, 361454720, 6652338176, 129341001216, 2645494627328, 56734280221696, 1272300911597568, 29769957834147840, 725430667245355008, 18379623419316338688, 483476314203202945024, 13187069277429966733312, 372512001057014648537088, 10886129458069912361631744, 328776894530826384975593472
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OFFSET
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1,3
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COMMENTS
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O.g.f. equals the logarithm of the e.g.f. of A300988.
The e.g.f. G(x) of A300988 satisfies: [x^n] G(x)^(4*n) = (n+3) * [x^(n-1)] G(x)^(4*n) for n>=1.
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LINKS
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FORMULA
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O.g.f. A(x) satisfies: [x^n] exp( 4*n * A(x) ) = (n + 3) * [x^(n-1)] exp( 4*n * A(x) ) for n>=1.
a(n) ~ c * n! * n^7, where c = 0.00000132855349... - Vaclav Kotesovec, Mar 20 2018
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EXAMPLE
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O.g.f.: A(x) = x + x^2 + 6*x^3 + 50*x^4 + 520*x^5 + 6312*x^6 + 86080*x^7 + 1288704*x^8 + 20862720*x^9 + 361454720*x^10 + ...
where
A(x) = x * (1 - 3*x*A'(x)) / (1 - 4*x*A'(x)).
RELATED SERIES.
exp(A(x)) = 1 + x + 3*x^2/2! + 43*x^3/3! + 1369*x^4/4! + 69561*x^5/5! + 4991371*x^6/6! + 471516403*x^7/7! + 56029153713*x^8/8! + 8112993527089*x^9/9! + ... + A300988(n)*x^n/n! + ...
A'(x) = 1 + 2*x + 18*x^2 + 200*x^3 + 2600*x^4 + 37872*x^5 + 602560*x^6 + 10309632*x^7 + 187764480*x^8 + 3614547200*x^9 + ...
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PROG
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(PARI) {a(n) = my(A=x); for(i=1, n, A = x*(1-3*x*A')/(1-4*x*A' +x*O(x^n))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
(PARI) /* [x^n] exp( 4*n * A(x) ) = (n + 3) * [x^(n-1)] exp( 4*n * A(x) ) */
{a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^(4*(#A-1))); A[#A] = ((#A+2)*V[#A-1] - V[#A])/(4*(#A-1)) ); polcoeff( log(Ser(A)), n)}
for(n=1, 25, 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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