A370437 Expansion of g.f. A(x) satisfying A( x*(1 + 2*x)*A(x) )^3 = A( x^2*(1 + 3*x)*A(x) )^2.

1, 0, -3, 10, -15, 84, -161, -174, 612, 1596, 1926, -38592, -5895, 234684, 603621, -2907414, -7559802, 15762420, 116671738, -95405184, -1326191061, -582470792, 14912006205, 21762765822, -142157118077, -418666482912, 1222704927801, 6161408393758, -7256289555369, -80123028681972

Offset

1,3

Links

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

Example

G.f.: A(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 + 603621*x^15 + ...
where A( x*(1 + 2*x)*A(x) )^3 = A( x^2*(1 + 3*x)*A(x) )^2.
RELATED SERIES.
B(x) = A( x*(1 + 2*x)*A(x) )^(1/2) = A( x^2*(1 + 3*x)*A(x) )^(1/3)
where B(x) is the g.f. of A370438, which begins
B(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 + ...
B(x)^2 = A( x*(1 + 2*x)*A(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 - 23640*x^13 - 51605*x^14 + 134174*x^15 + ...
B(x)^3 = A( x^2*(1 + 3*x)*A(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 - 33825*x^14 - 124742*x^15 + 217842*x^16 + ...
B(x)^6 = A( x*(1 + 2*x)*A(x) )^3 = 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 - 61476*x^17 - 530776*x^18 + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); Ax = x*Ser(A);
A[#A] = polcoeff( subst(Ax, x, x^2*(1 + 3*x)*Ax )^2 - subst(Ax, x, x*(1 + 2*x)*Ax )^3, #A+5); ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A370438, A370537 (dual), A370535.

Sequence in context: A037345 A217278 A175336 * A259877 A182334 A051420

Adjacent sequences: A370434 A370435 A370436 * A370438 A370439 A370440

Keyword

sign

Author

Paul D. Hanna, Mar 07 2024

Status

approved