A089796 Series reversion of g.f. A(x) is -A(-x).

1, 1, 1, 2, 4, 8, 17, 38, 87, 203, 482, 1160, 2822, 6929, 17149, 42736, 107144, 270060, 683940, 1739511, 4441255, 11378814, 29245927, 75386341, 194838673, 504802508, 1310843123, 3411070838, 8893590454, 23230151872, 60780378423, 159281034709

Offset

1,4

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..550

Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.

Formula

G.f. A(x)=y satisfies (x-x^2-x^3)+y*(-1+3*x-x^3)+y^2*(-1+x^2-x^3)+y^3*(1-x+x^2-x^3)=0.

a(n) ~ sqrt((-1 - 3*s + s^3 - 2*r*(-1 + s^2 + s^3) + 3*r^2*(1 + s + s^2 + s^3)) / (1 - 3*s + 3*r*s - r^2*(1 + 3*s) + r^3*(1 + 3*s))) / (2*sqrt(Pi) * n^(3/2) * r^(n-1/2)), where r = 0.3637032853572807454... and s = 0.8232781794881572707... are roots of the system of equations 3*r = 1 + r^3 + 2*(1 - r^2 + r^3)*s + 3*(-1 + r - r^2 + r^3)*s^2, s + s^2 + r*(-1 - 3*s + s^3) + r^3*(1 + s + s^2 + s^3) = s^3 + r^2*(-1 + s^2 + s^3). - Vaclav Kotesovec, Mar 10 2014

Mathematica

terms = 32; A[_] = 0;
Do[A[x_] = (x-x^2-x^3) + (3x-x^3) A[x] + (-1+x^2-x^3) A[x]^2 + (1-x+x^2-x^3) A[x]^3 + O[x]^(terms+1) // Normal, {terms+1}];
CoefficientList[A[x]/x, x] (* Jean-François Alcover, Nov 10 2018 *)

Prog

(PARI) a(n)=local(A); if(n<1, 0, A=O(x); for(k=1, n, A=(x-x^2-x^3)+A*(3*x-x^3)+A^2*(-1+x^2-x^3)+A^3*(1-x+x^2-x^3)); polcoeff(A, n))

Crossrefs

a(n) = A006196(n) if n<28.

Sequence in context: A154222 A114199 A006196 * A112482 A193050 A107597

Adjacent sequences: A089793 A089794 A089795 * A089797 A089798 A089799

Keyword

nonn

Author

Michael Somos, Nov 10 2003

Status

approved