A370536 Expansion of g.f. A(x) = G( x^2*(1 + 3*x)*G(x) )^(1/3) = G( x^3*(1 + 5*x)*G(x)^2 )^(1/5), where G(x) is the g.f. of A370535.

1, 1, 4, -20, 100, -500, 2530, -12290, 63970, -310770, 1580415, -7901235, 39580710, -199089900, 1000497720, -5029848484, 25374291371, -127907347261, 646677792365, -3273908647705, 16606252541363, -84435428096707, 430294257188507, -2198480870662015, 11264938277198125

Offset

1,3

Links

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

Example

G.f.: A(X) = x + x^2 + 4*x^3 - 20*x^4 + 100*x^5 - 500*x^6 + 2530*x^7 - 12290*x^8 + 63970*x^9 - 310770*x^10 + 1580415*x^11 - 7901235*x^12 + ...
where A(x) = G( x^2*(1 + 3*x)*G(x) )^(1/3) = G( x^3*(1 + 5*x)*G(x)^2 )^(1/5)
and G(x) is the g.f. of A370535, which begins
G(x) = x + 15*x^3 - 80*x^4 + 480*x^5 - 2832*x^6 + 16555*x^7 - 94350*x^8 + 544050*x^9 - 3048150*x^10 + 17177355*x^11 - 95672700*x^12 + ...
RELATED SERIES.
A(x)^3 = G( x^2*(1 + 3*x)*G(x) ) = x^3 + 3*x^4 + 15*x^5 - 35*x^6 + 240*x^7 - 1392*x^8 + 8074*x^9 - 44550*x^10 + 262080*x^11 - 1413200*x^12 + ...
A(x)^5 = G( x^3*(1 + 5*x)*G(x)^2 ) = x^5 + 5*x^6 + 30*x^7 - 10*x^8 + 385*x^9 - 2139*x^10 + 13590*x^11 - 80910*x^12 + 515970*x^13 - 2952970*x^14 + ...
A(x)^15 = x^15 + 15*x^16 + 165*x^17 + 995*x^18 + 5805*x^19 + 16083*x^20 + 93075*x^21 - 82575*x^22 + 2166975*x^23 - 11141575*x^24 + 99995160*x^25 + ...
where A(x)^15 = G( x^2*(1 + 3*x)*G(x) )^5 = G( x^3*(1 + 5*x)*G(x)^2 )^3.

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 )^5 - subst(G, x, x^3*(1 + 5*x)*G^2 )^3, #V+14); ); A = subst(G, x, x^2*(1 + 3*x)*G )^(1/3); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A370535, A370534, A370438, A370538.

Sequence in context: A178874 A103771 A005054 * A216099 A105480 A242156

Adjacent sequences: A370533 A370534 A370535 * A370537 A370538 A370539

Keyword

sign

Author

Paul D. Hanna, Mar 08 2024

Status

approved