A274569 G.f. satisfies: A( x*A(x) ) = x^2 + x^3.

1, 1, -1, -2, 2, 9, 1, -29, -24, 88, 153, -234, -796, 327, 3509, 1301, -13924, -16511, 47366, 109639, -121886, -583921, 79027, 2691465, 1808431, -10775705, -16965876, 35686874, 107103382, -77820607, -560120299, -95094380, 2536045800, 2521002564, -9832805334, -19928899203, 29983217002, 118838292930, -44109504096, -600237397739, -285632368107, 2622720919077, 3530864633371, -9611558966277, -24857829812388, 25472208656701, 140445352832736, -5891793579597, -680872708032455, -537359624615144, 2838471301330439, 4957200244969051, -9600812793387365

Offset

1,4

Links

Paul D. Hanna, Table of n, a(n) for n = 1..500

Example

G.f.: A(x) = x + x^2 - x^3 - 2*x^4 + 2*x^5 + 9*x^6 + x^7 - 29*x^8 - 24*x^9 + 88*x^10 + 153*x^11 - 234*x^12 - 796*x^13 + 327*x^14 + 3509*x^15 + 1301*x^16 +...
such that A( x*A(x) ) = x^2 + x^3.

Maple

N:= 50: # to get a(1) to a(N)
a[1]:= 1:
eq:= eval(A(x*A(x)) - x^2 - x^3, A = unapply(add(a[i]*x^i, i=1..N), x)):
S:= map(normal, series(eq, x, N+2)):
for n from 2 to N+1 do
  a[n]:= solve(coeff(S, x, n+1))
od:
seq(a[i], i=1..50); # Robert Israel, Jul 26 2016

Mathematica

nmax = 53; sol = {a[1] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 1, n}] /. sol; eq = CoefficientList[A[x A[x]] - (x^2 + x^3) + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax+1}];
sol /. Rule -> Set;
a /@ Range[1, nmax] (* Jean-François Alcover, Nov 01 2019 *)

Prog

(PARI) {a(n) = my(A=[1, 1], F); for(i=1, n, A=concat(A, 0); F=x*Ser(A); F = subst(F, x, x*F); A[#A] = -Vec(F)[#A]); A[n]}
for(n=1, 60, print1(a(n), ", "))

Crossrefs

Sequence in context: A048650 A125313 A199058 * A082838 A074961 A359454

Adjacent sequences: A274566 A274567 A274568 * A274570 A274571 A274572

Keyword

sign

Author

Paul D. Hanna, Jul 18 2016

Status

approved