|
Search: id:A075834
|
|
|
| A075834 |
|
Coefficients of power series A(x) such that n-th term of A(x)^n = n! x^(n-1) for n>0. |
|
+0
16
|
|
| 1, 1, 1, 2, 7, 34, 206, 1476, 12123, 111866, 1143554, 12816572, 156217782, 2057246164, 29111150620, 440565923336, 7101696260883, 121489909224618, 2198572792193786, 41966290373704332, 842706170872913634
(list; graph; listen)
|
|
|
OFFSET
|
0,4
|
|
|
COMMENT
|
Also, number of stablized-interval-free permutations on [n] (see Callan link).
|
|
LINKS
|
David Callan, Counting stabilized-interval-free permutations
David Callan, Counting Stabilized-Interval-Free Permutations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.1.8.
|
|
FORMULA
|
a(0)=a(1)=1, a(n)=(n-1)*a(n-1) + sum_{j=2..n-2}(j-1)*a(j)*a(n-j), n>=2 (from David Callan).
G.f.: A(x) = x/series_reversion(x*G(x)); G(x) = A(x*G(x)); A(x) = G(x/A(x)); where G(x) is the g.f. of the factorials (A000142). - Paul D. Hanna (pauldhanna(AT)juno.com), Jul 09 2006
G.f.: A(x) = 1 + x/(1 - x*A'(x)/A(x)) = 1 + x/(1-x - x^2*d/dx[(A(x) - 1)/x)]).
G.f.: A(x) = 1 + x*F(x) where F(x) satisfies F(x) = 1 + x*F(x) + x^2*F(x)*F'(x) and F'(x) = d/dx F(x). [From Paul D. Hanna (pauldhanna(AT)juno.com), Sep 02 2008]
|
|
EXAMPLE
|
At n=7, the 7-th term of A(x)^7 is 7! x^6, as demonstrated by A(x)^7 = 1 + 7 x + 28 x^2 + 91 x^3 + 294 x^4 + 1092 x^5 + 5040 x^6 + 29093 x^7 + 203651 x^8 + ...
A(x) = 1 + x + x^2 + 2*x^3 + 7*x^4 + 34*x^5 + 206*x^6 +... = x/series_reversion(x + x^2 + 2*x^3 + 6*x^4 + 24*x^5 + 120*x^6 +...).
Related expansions:
log(A(x)) = x + x^2/2 + 4*x^3/3 + 21*x^4/4 + 136*x^5/5 + 1030*x^6/6 +..;
1 - x/(A(x) - 1) = x + x^2 + 4*x^3 + 21*x^4 + 136*x^5 + 1030*x^6 +..;
derivative[(A(x) - 1)/x] = 1 + 4*x + 21*x^2 + 136*x^3 + 1030*x^4 +...
|
|
PROGRAM
|
(PARI) a(n)=if(n<0, 0, if(n<=1, 1, (n-1)*a(n-1)+sum(j=2, n-2, (j-1)*a(j)*a(n-j)); ))
(PARI) a(n)=Vec(x/serreverse(x*Ser(vector(n+1, k, (k-1)!))))[n+1] - Paul D. Hanna (pauldhanna(AT)juno.com), Jul 09 2006
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1+x/(1-x*deriv(A)/A)); polcoeff(A, n)}
(PARI) {a(n)=local(F=1+x*O(x^n)); for(i=0, n, F=1+x*F+x^2*F*deriv(F)+x*O(x^n)); polcoeff(1+x*F, n)} [From Paul D. Hanna (pauldhanna(AT)juno.com), Sep 02 2008]
|
|
CROSSREFS
|
Cf. A091063, A084938.
Sequence in context: A117399 A145345 A056543 this_sequence A011800 A112916 A145845
Adjacent sequences: A075831 A075832 A075833 this_sequence A075835 A075836 A075837
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Paul D. Hanna (pauldhanna(AT)juno.com), Oct 14 2002, Jul 30 2008
|
|
EXTENSIONS
|
More terms from David Wasserman (dwasserm(AT)earthlink.net), Jan 26 2005
|
|
|
Search completed in 0.034 seconds
|
