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A268655
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G.f. A(x) satisfies: A( A(x)^2 - A(x)^3 ) = x*A(x).
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2
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1, 1, 1, 2, 6, 17, 45, 123, 356, 1061, 3193, 9691, 29741, 92228, 288331, 907316, 2871818, 9138352, 29217870, 93813839, 302367222, 977913351, 3172727729, 10323213290, 33677758165, 110135406578, 360982631871, 1185632194394, 3901702645568, 12862978506084, 42477650804971, 140496123308659, 465383254510707, 1543693890316144
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OFFSET
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1,4
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COMMENTS
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Compare g.f. to: C( C(x)^2 - C(x)^4 ) = C(x)^2 and C( C(x) - C(x)^2 ) = C(x) where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
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LINKS
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FORMULA
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G.f. A(x) satisfies: A( A(x^2-x^3)/x ) = x.
a(n) ~ c * d^n / n^(3/2), where d = 3.469666051910765293806107674644687565... and c = 0.1297182904218030477707168121291985... . - Vaclav Kotesovec, Apr 01 2016
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EXAMPLE
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G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 6*x^5 + 17*x^6 + 45*x^7 + 123*x^8 + 356*x^9 + 1061*x^10 + 3193*x^11 + 9691*x^12 + 29741*x^13 + 92228*x^14 + 288331*x^15 +...
such that A( A(x)^2 - A(x)^3 ) = x*A(x).
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 17*x^6 + 50*x^7 + 140*x^8 + 394*x^9 + 1152*x^10 + 3464*x^11 + 10541*x^12 + 32320*x^13 + 99973*x^14 +...
A(x)^3 = x^3 + 3*x^4 + 6*x^5 + 13*x^6 + 36*x^7 + 108*x^8 + 316*x^9 + 918*x^10 + 2727*x^11 + 8291*x^12 + 25521*x^13 + 79110*x^14 + 246942*x^15 +...
A(x)^2 - A(x)^3 = x^2 + x^3 + 4*x^6 + 14*x^7 + 32*x^8 + 78*x^9 + 234*x^10 + 737*x^11 + 2250*x^12 + 6799*x^13 + 20863*x^14 + 65056*x^15 +...
The series reversion of g.f. A(x) equals A(x^2-x^3)/x, which begins:
A(x^2-x^3)/x = x - x^2 + x^3 - 2*x^4 + 2*x^5 - 3*x^6 + 5*x^7 - 9*x^8 + 18*x^9 - 38*x^10 + 79*x^11 - 162*x^12 + 330*x^13 - 661*x^14 + 1323*x^15 - 2661*x^16 + 5392*x^17 - 11037*x^18 + 22802*x^19 - 47447*x^20 + 99238*x^21 - 208283*x^22 +...
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PROG
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(PARI) /* From definition A( A(x)^2 - A(x)^3 ) = x*A(x) */
{a(n) = my(A=[1, 1]); for(i=1, n, A = concat(A, 0); G=x*Ser(A); A[#A] = -Vec(subst(G, x, G^2 - G^3 ))[#A] ); A[n]}
for(n=1, 40, print1(a(n), ", "))
(PARI) /* Using Series Reversion of A(x^2-x^3)/x */
{a(n) = my(A=x); for(i=1, n, A = serreverse( subst(A, x, x^2 - x^3 +x^2*O(x^n))/x) ); 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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