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URL: https://oeis.org/A173895

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A173895
E.g.f. satisfies: A'(x) = 1/(1 + x*A(x)) with A(0) = 1.
3
1, 1, -1, 0, 9, -48, 15, 2448, -24927, 23424, 3091311, -47659200, 88056969, 10702667520, -225139993377, 679791291648, 78646340795265, -2128005345251328, 9456106738649631, 1053535684549174272
OFFSET
0,5
COMMENTS
Define a polynomial sequence P_n(x) recursively by
... P_0(x) = 1, and for n >= 1
... P_n(x) = (x-1)*P_(n-1)(x-1)-n*P_(n-1)(x+1).
The first few polynomials are
P_1(x) = x-2
P_2(x) = x^2-6*x+5
P_3(x) = x^3-12*x^2+32*x-12.
It appears that a(n+1) = P_n(1) (checked as far as a(19)).
Compare with A144010.
LINKS
FORMULA
E.g.f. satisfies: A(x) = 1 + Integral 1/(1 + x*A(x)) dx.
E.g.f. satisfies: A(G(x)) = 1 + x where G(x) is the e.g.f. of A000932 (offset 1). [Paul D. Hanna, Aug 23 2011]
EXAMPLE
E.g.f.: A(x) = 1 + x - x^2/2! + 9*x^4/4! - 48*x^5/5! + 15*x^6/6! + 2448*x^7/7! +...
where
1/(1 + x*A(x)) = 1 - x + 9*x^3/3! - 48*x^4/4! + 15*x^5/5! + 2448*x^6/6! +...
Also, A(G(x)) = 1 + x where
G(x) = x + x^2/2! + 3*x^3/3! + 6*x^4/4! + 18*x^5/5! + 48*x^6/6! + 156*x^7/7! + 492*x^8/8! +...+ A000932(n-1)*x^n/n! +...
MATHEMATICA
m = 20; A[_] = 1;
Do[A[x_] = 1 + Integrate[1/(1+x*A[x])+O[x]^m, x]+O[x]^m // Normal, {m}];
CoefficientList[A[x], x] * Range[0, m-1]! (* Jean-François Alcover, Nov 02 2019 *)
PROG
(PARI) {a(n)=local(A=1+x); for(i=0, n, A=1+intformal(1/(1+x*A+x*O(x^n)) )); n!*polcoeff(A, n)}
CROSSREFS
Sequence in context: A207318 A293042 A159525 * A392338 A341757 A380964
KEYWORD
easy,sign
AUTHOR
Peter Bala, Nov 26 2010
STATUS
approved