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A370538
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Expansion of g.f. A(x) = G( x*(1 + 2*x)*G(x) )^(1/2) = G( x*(1 + 3*x)*G(x)^2 )^(1/3), where G(x) is the g.f. of A370537.
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5
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1, 1, 1, -3, 15, 3, 148, -314, 466, -1980, 13410, -12348, 52579, -312347, 898033, -221141, 10087231, -27550253, 35879866, -120326612, 1290473218, -1015145068, 2822763419, -31671326569, 92976002320, 40883035636, 903217721062, -3628100989426, 1532499458666, -6655463860036
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OFFSET
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1,4
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LINKS
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EXAMPLE
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G.f.: A(x) = x + x^2 + x^3 - 3*x^4 + 15*x^5 + 3*x^6 + 148*x^7 - 314*x^8 + 466*x^9 - 1980*x^10 + 13410*x^11 - 12348*x^12 + 52579*x^13 - 312347*x^14 + 898033*x^15 + ...
where A(x) = G( x*(1 + 2*x)*G(x) )^(1/2) = G( x*(1 + 3*x)*G(x)^2 )^(1/3),
and G(x) is the g.f. of A370537, which begins
G(x) = x + 3*x^3 - 10*x^4 + 42*x^5 - 72*x^6 + 432*x^7 - 1296*x^8 + 3474*x^9 - 11644*x^10 + 48438*x^11 - 119532*x^12 + 385150*x^13 + ...
RELATED SERIES.
A(x)^2 = G( x*(1 + 2*x)*G(x) ) = x^2 + 2*x^3 + 3*x^4 - 4*x^5 + 25*x^6 + 30*x^7 + 341*x^8 - 416*x^9 + 807*x^10 - 4454*x^11 + 30125*x^12 + ...
A(x)^3 = G( x*(1 + 3*x)*G(x)^2 ) = x^3 + 3*x^4 + 6*x^5 - 2*x^6 + 33*x^7 + 75*x^8 + 607*x^9 - 189*x^10 + 1287*x^11 - 7143*x^12 + 48735*x^13 + ...
A(x)^6 = x^6 + 6*x^7 + 21*x^8 + 32*x^9 + 90*x^10 + 324*x^11 + 2064*x^12 + 4032*x^13 + 9513*x^14 - 6310*x^15 + 116499*x^16 + ...
where A(x)^6 = G( x*(1 + 2*x)*G(x) )^3 = G( x*(1 + 3*x)*G(x)^2 )^2.
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PROG
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(PARI) {a(n) = my(A, G, V=[1]); for(i=1, n+1, V = concat(V, 0); G = x*Ser(V);
V[#V] = polcoeff( subst(G, x, x*(1 + 2*x)*G )^3 - subst(G, x, x*(1 + 3*x)*G^2 )^2, #V+5); ); A = subst(G, x, x*(1 + 2*x)*G )^(1/2); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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