A212919 G.f. satisfies: A(x) = x^3 - x + Series_Reversion(x - x*A(x)).

1, 1, 1, 1, 5, 14, 29, 73, 229, 671, 1840, 5415, 16983, 52547, 161420, 511039, 1655598, 5372395, 17527912, 58076084, 194676024, 656160449, 2227549164, 7635624954, 26380508479, 91696805060, 320866223000, 1130833326852, 4010720214072, 14306769257286

Offset

3,5

Comments

Compare the g.f. to a g.f. G(x) of A088714 (offset 1), which satisfies:

G(x) = Series_Reversion(x - x*G(x)),

G(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*G(x)^n/n!, and

G(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*G(x)^n/n! ).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 3..200

Formula

G.f. A(x) also satisfies:

(1) A(x) = x^3 + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*A(x)^n/n!.

(2) A(x) = x^3 - x + x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*A(x)^n/n! ).

Example

G.f.: A(x) = x^3 + x^4 + x^5 + x^6 + 5*x^7 + 14*x^8 + 29*x^9 + 73*x^10 +...
The series reversion of x - x*A(x) begins:
x + x^4 + x^5 + x^6 + 5*x^7 + 14*x^8 + 29*x^9 + 73*x^10 + 229*x^11 +...
which equals x - x^3 + A(x).
The g.f. satisfies:
A(x) = x^3 + x*A(x) + d/dx x^2*A(x)^2/2! + d^2/dx^2 x^3*A(x)^3/3! + d^3/dx^3 x^4*A(x)^4/4! +...
log(1-x^2 + A(x)/x) = A(x) + d/dx x*A(x)^2/2! + d^2/dx^2 x^2*A(x)^3/3! + d^3/dx^3 x^3*A(x)^4/4! +...
Related expansions:
d/dx x^2*A(x)^2/2! = 4*x^7 + 9*x^8 + 15*x^9 + 22*x^10 + 78*x^11 + 260*x^12 +...
d^2/dx^2 x^3*A(x)^3/3! = 22*x^10 + 78*x^11 + 182*x^12 + 350*x^13 + 1080*x^14 +...
d^3/dx^3 x^4*A(x)^4/4! = 140*x^13 + 680*x^14 + 2040*x^15 + 4845*x^16 +...
d^4/dx^4 x^5*A(x)^5/5! = 969*x^16 + 5985*x^17 + 21945*x^18 + 61985*x^19 +...
...
d^(n-1)/dx^(n-1) x^n*A(x)^n/n! = A002293(n)*x^(3*n+1) +...

Prog

(PARI) {a(n)=local(A=x^3); for(i=1, n, A=x^3-x+serreverse(x-x*A +x*O(x^n))); polcoeff(A, n)}
for(n=3, 40, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(G=F); for(i=1, n, G=deriv(G)); G}
{a(n)=local(A=x^3); for(i=1, n, A=x^3+sum(m=1, n, Dx(m-1, x^m*A^m/m!)+x*O(x^n))); polcoeff(A, n)}
for(n=3, 40, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(G=F); for(i=1, n, G=deriv(G)); G}
{a(n)=local(A=x^3); for(i=1, n, A=x^3-x+x*exp(sum(m=1, n, Dx(m-1, x^(m-1)*A^m/m!)+x*O(x^n)))); polcoeff(A, n)}
for(n=3, 40, print1(a(n), ", "))

Crossrefs

Cf. A088714, A212910.

Sequence in context: A374929 A299291 A019262 * A280230 A076042 A231669

Adjacent sequences: A212916 A212917 A212918 * A212920 A212921 A212922

Keyword

nonn

Author

Paul D. Hanna, May 31 2012

Status

approved