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A295808
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G.f. A(x) satisfies: A(x)^3 = 1+x + x*(A(x)^4 + A(x)^8).
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3
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1, 1, 3, 17, 110, 783, 5908, 46433, 376029, 3115941, 26293410, 225166050, 1951877304, 17094430060, 151026790086, 1344405191931, 12046557221374, 108569555036541, 983512740523989, 8950335155129326, 81786796816686222, 750133309106091800, 6903286130025559800, 63724450582843480092, 589897018021520290940, 5474784437150040712036, 50932035767512193052753, 474865449870890392910894
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OFFSET
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0,3
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COMMENTS
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Compare to: G(x)^3 = 1+x + x*(G(x)^2 + G(x)^4) holds when G(x) = (1 + x - sqrt(1 - 2*x - 3*x^2))/(2*x) is a g.f. of the Motzkin numbers (A001006).
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LINKS
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FORMULA
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G.f. A(x) satisfies:
(1) A(x) = 1 + Series_Reversion( x/(1 + 3*x + 8*x^2 + 11*x^3 + 10*x^4 + 5*x^5 + x^6) ).
(2) F(A(x)) = x such that F(x) = -(1-x)/(1 - x + x^3 - x^5 + x^6).
(3) A(x) satisfies: A(x) = 1/A(-x*A(x)^5).
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EXAMPLE
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G.f.: A(x) = 1 + x + 3*x^2 + 17*x^3 + 110*x^4 + 783*x^5 + 5908*x^6 + 46433*x^7 + 376029*x^8 + 3115941*x^9 + 26293410*x^10 + 225166050*x^11 + 1951877304*x^12 +...
such that A(x)^3 = 1+x + x*(A(x)^4 + A(x)^8).
RELATED SERIES.
A(x)^3 = 1 + 3*x + 12*x^2 + 70*x^3 + 468*x^4 + 3393*x^5 + 25932*x^6 + 205716*x^7 + 1677804*x^8 + 13980710*x^9 + 118505772*x^10 +...
A(x)^4 = 1 + 4*x + 18*x^2 + 108*x^3 + 735*x^4 + 5388*x^5 + 41496*x^6 + 331036*x^7 + 2711511*x^8 + 22670964*x^9 + 192695140*x^10 +...
A(x)^8 = 1 + 8*x + 52*x^2 + 360*x^3 + 2658*x^4 + 20544*x^5 + 164220*x^6 + 1346768*x^7 + 11269199*x^8 + 95834808*x^9 + 825905828*x^10 +...
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PROG
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(PARI) {a(n) = my(A=1+x); for(i=1, n, A = ((1+x) + x*A^4 + x*A^8 +x*O(x^n))^(1/3) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=1+x); A = 1 + serreverse( x/(1 + 3*x + 8*x^2 + 11*x^3 + 10*x^4 + 5*x^5 + x^6 +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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