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A228987
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G.f. satisfies A(x) = 1 + x*A(x)^3 + x^3*A(x)^8.
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1
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1, 1, 3, 13, 66, 364, 2116, 12768, 79222, 502297, 3240120, 21196593, 140295584, 937787728, 6321624862, 42926227470, 293350136170, 2015999854478, 13923926272607, 96598395025615, 672852440805930, 4703751150849738, 32991286060134402, 232091541493091566
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OFFSET
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0,3
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LINKS
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FORMULA
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G.f. satisfies: A(x) = 1/A(-x*A(x)^5); note that the function G(x) = 1 + x*G(x)^3 (g.f. of A001764) also satisfies this condition: G(x) = 1/G(-x*G(x)^5).
a(n) ~ sqrt((3 - r*s^2)/(2*Pi*(3 + 28*r^2*s^5))) / (4*n^(3/2)*r^(n + 1/2)), where r = 0.1331154541373089587498695338172936885734070972340... and s = 1.408602671059676188189711196409966797670750551605... are real roots of the system of equations 1 + r*s^3 + r^3*s^8 = s, 3*r*s^2 + 8*r^3*s^7 = 1. - Vaclav Kotesovec, Nov 22 2017
a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k,k) * binomial(3*n-k+1,n-2*k)/(3*n-k+1). - Seiichi Manyama, Aug 28 2023
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EXAMPLE
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G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 66*x^4 + 364*x^5 + 2116*x^6 +...
Related expansions:
A(x)^3 = 1 + 3*x + 12*x^2 + 58*x^3 + 312*x^4 + 1788*x^5 + 10686*x^6 +...
A(x)^8 = 1 + 8*x + 52*x^2 + 328*x^3 + 2082*x^4 + 13384*x^5 + 87124*x^6 +...
Given (1) A(x) = 1 + x*A(x)^3 + x^3*A(x)^8,
suppose (2) A(x) = 1/A(-x*A(x)^5),
then substituting x in (1) with -x*A(x)^5 yields:
1/A(x) = 1 - x*A(x)^5/A(x)^3 - x^3*A(x)^15/A(x)^8,
which illustrates that (2) is consistent with (1).
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PROG
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(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*A^3+x^3*A^8 +x*O(x^n)); 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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