A158884 G.f. A(x) satisfies: d/dx x*A(x) = 1+x + x*[d/dx log(A(x))].

1, 1, -1, 4, -23, 166, -1410, 13602, -145803, 1711690, -21785618, 298370920, -4372151566, 68234087624, -1129894265272, 19788479904366, -365520041466291, 7103187300763530, -144897616964143050, 3096285550330959336

Offset

0,4

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Formula

G.f. satisfies: x*A'(x) = A(x)*(1+x - A(x))/(A(x) - 1).

G.f.: A(x) = 1/G(-x) where G(x) is the g.f. of A088715.

G.f. satisfies: A(x/F(x)) = F(x) where F(x) is the g.f. of A158883.

G.f. satisfies: A(x*H(-x)) = H(-x) where H(x) is the g.f. of A088716.

G.f. satisfies: [x^n] 1/A(-x)^(n+2) = [x^(n+1)] 1/A(-x)^(n+2)/(n+2) = A088716(n+1).

a(n) ~ -(-1)^n * c * n! * n^2, where c = A238223 / exp(1) = 0.080179614624692622... - Vaclav Kotesovec, Nov 21 2017

Example

G.f.: A(x) = 1 + x - x^2 + 4*x^3 - 23*x^4 + 166*x^5 - 1410*x^6 +...
d/dx x*A(x) = 1 + 2*x - 3*x^2 + 16*x^3 - 115*x^4 + 996*x^5 - 9870*x^6 +...
d/dx log(A(x)) = 1 - 3*x + 16*x^2 - 115*x^3 + 996*x^4 - 9870*x^5 +...
Coefficients in powers A(x)^-n begin:
A(x)^-1: (1),-1,2,-7,36,-240,1926,-17815,184916,...;
A(x)^-2: (1),(-2),5,-18,90,-580,4525,-40946,417822,...;
A(x)^-3: 1,(-3),(9),-34,168,-1053,7997,-70776,709614,...;
A(x)^-4: 1,-4,(14),(-56),277,-1700,12594,-109032,1073658,...;
A(x)^-5: 1,-5,20,(-85),(425),-2571,18630,-157860,1526330,...;
A(x)^-6: 1,-6,27,-122,(621),(-3726),26492,-219912,2087658,...;
A(x)^-7: 1,-7,35,-168,875,(-5236),(36652),-298446,2782080,...;
A(x)^-8: 1,-8,44,-224,1198,-7184,(49680),(-397440),3639333,...; ...
where coefficients in parenthesis form A158883 and signed A088716
and A(x)^-1 (first row) is the g.f. of signed A088715.

Prog

(PARI) {a(n)=local(A=[1, 1]); for(i=2, n, A=concat(A, 0); A[ #A]=(Vec(Ser(A)^(#A-1))-Vec(Ser(A)^(#A)))[ #A]); Vec(Ser(A)^(n+1)/(n+1))[n+1]}

Crossrefs

Cf. A158883, A088715, A088716.

Sequence in context: A354497 A111547 A171992 * A317057 A053525 A277382

Adjacent sequences: A158881 A158882 A158883 * A158885 A158886 A158887

Keyword

sign

Author

Paul D. Hanna, Apr 30 2009

Status

approved