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A273320
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G.f. A(x) satisfies: A(x)^2 - 2*A(x)^3 = -A(-x^2).
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1
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1, 1, 2, 7, 26, 98, 389, 1617, 6884, 29818, 131284, 585966, 2644084, 12041541, 55280357, 255556439, 1188632292, 5558332388, 26116960640, 123243816854, 583830832350, 2775426824820, 13235989547850, 63306397853250, 303597924249840, 1459546439694004, 7032721684219584, 33958054240107363, 164290025822878210, 796288552943558821, 3866056901078138147, 18799970286124092609, 91557579962243322708
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OFFSET
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1,3
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LINKS
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FORMULA
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G.f. A(x) satisfies: -A(-B(x)^2) = x^2 - 2*x^3, where A(B(x)) = x.
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EXAMPLE
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G.f.: A(x) = x + x^2 + 2*x^3 + 7*x^4 + 26*x^5 + 98*x^6 + 389*x^7 + 1617*x^8 + 6884*x^9 + 29818*x^10 + 131284*x^11 + 585966*x^12 +...
where A(x)^2 - 2*A(x)^3 = -A(-x^2).
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 5*x^4 + 18*x^5 + 70*x^6 + 276*x^7 + 1127*x^8 + 4768*x^9 + 20606*x^10 + 90414*x^11 + 402210*x^12 + 1810476*x^13 +...
A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 34*x^6 + 138*x^7 + 567*x^8 + 2384*x^9 + 10290*x^10 + 45207*x^11 + 201154*x^12 + 905238*x^13 +...
where
A(x)^2 - 2*A(x)^3 = x^2 - x^4 + 2*x^6 - 7*x^8 + 26*x^10 - 98*x^12 + 389*x^14 - 1617*x^16 + 6884*x^18 - 29818*x^20 + 131284*x^22 - 585966*x^24 +...
Let B(x) be the series reversion of g.f. A(x), so that A(B(x)) = x, then
B(x) = x - x^2 - 2*x^4 - x^7 - 9*x^8 + 15*x^9 - 19*x^10 - 21*x^11 + 9*x^12 - 30*x^13 - 74*x^14 - 119*x^15 + 5*x^16 - 216*x^17 - 2164*x^18 + 3937*x^19 - 10603*x^20 + 9568*x^21 - 26632*x^22 + 22777*x^23 - 63015*x^24 - 42449*x^25 + 69029*x^26 +...
where A(-B(x)^2) = 2*x^3 - x^2,
also, B(2*x^3 - x^2) = -B(x)^2.
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PROG
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(PARI) /* From -A(-B(x)^2) = x^2 - 2*x^3, where A(B(x)) = x: */
{a(n) = my(A=[1, 1], F, B); for(i=1, n, A=concat(A, 0); F=x*Ser(A); B=serreverse(F); A[#A] = Vec(subst(-F, x, -B^2))[#A]/2); 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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