A370438 Expansion of g.f. A(x) = G( x*(1 + 2*x)*G(x) )^(1/2) = G( x^2*(1 + 3*x)*G(x) )^(1/3), where G(x) is the g.f. of A370437.

1, 1, -2, 4, -5, 31, -45, -57, -66, 1124, 116, -8314, -21328, 76424, 229013, -537885, -3308391, 2165589, 36968501, 16926113, -402914677, -602853397, 3854818970, 11099464628, -31321810278, -161502827466, 188659370973, 2076430792115, 118544397194, -23981088656686

Offset

1,3

Links

Paul D. Hanna, Table of n, a(n) for n = 1..501

Example

G.f.: A(x) = x + x^2 - 2*x^3 + 4*x^4 - 5*x^5 + 31*x^6 - 45*x^7 - 57*x^8 - 66*x^9 + 1124*x^10 + 116*x^11 - 8314*x^12 - 21328*x^13 + 76424*x^14 + 229013*x^15 + ...
where A(x) = G( x*(1 + 2*x)*G(x) )^(1/2) = G( x^2*(1 + 3*x)*G(x) )^(1/3)
and G(x) is the g.f. of  A370437, which begins
G(x) = x - 3*x^3 + 10*x^4 - 15*x^5 + 84*x^6 - 161*x^7 - 174*x^8 + 612*x^9 + 1596*x^10 + 1926*x^11 - 38592*x^12 - 5895*x^13 + 234684*x^14 + ...
RELATED SERIES.
A(x)^2 = G( x*(1 + 2*x)*G(x) ) = x^2 + 2*x^3 - 3*x^4 + 4*x^5 + 2*x^6 + 36*x^7 + 8*x^8 - 368*x^9 + 207*x^10 + 1674*x^11 + 3699*x^12 + ...
A(x)^3 = G( x^2*(1 + 3*x)*G(x) ) = x^3 + 3*x^4 - 3*x^5 + x^6 + 15*x^7 + 39*x^8 + 88*x^9 - 684*x^10 + 36*x^11 + 3514*x^12 + 6807*x^13 + ...
A(x)^6 = x^6 + 6*x^7 + 3*x^8 - 16*x^9 + 45*x^10 + 162*x^11 + 321*x^12 - 1044*x^13 - 4257*x^14 + 12694*x^15 + 37275*x^16 + ...
where A(x)^6 = G( x*(1 + 2*x)*G(x) )^3 = G( x^2*(1 + 3*x)*G(x) )^2.

Prog

(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^2*(1 + 3*x)*G )^2 - subst(G, x, x*(1 + 2*x)*G )^3, #V+5); ); A = subst(G, x, x*(1 + 2*x)*G )^(1/2); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A370437, A370538.

Sequence in context: A199929 A126666 A036983 * A303006 A154775 A210418

Adjacent sequences: A370435 A370436 A370437 * A370439 A370440 A370441

Keyword

sign

Author

Paul D. Hanna, Mar 08 2024

Status

approved