A355869 G.f. A(x) satisfies: x = Sum_{n=-oo..+oo} (-x)^n * (1 + A(x)*x^n)^n.

1, 1, 3, 8, 26, 69, 186, 554, 1606, 4772, 14825, 45920, 143586, 455561, 1446777, 4622609, 14885492, 48099814, 156175165, 509527438, 1667502213, 5475430124, 18036181574, 59562083008, 197197284543, 654439115129, 2176448986396, 7252784407286, 24214655260997

Offset

0,3

Comments

Related identity: Sum_{n=-oo..+oo} (-x)^n * (x^n + y)^n = 0 for all y.

Related identity: Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + y*x^n)^n = 0 for all y.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Formula

G.f. A(x) satisfies:

(1) x = Sum_{n=-oo..+oo} (-x)^n * (1 + A(x)*x^n)^n.

(2) 0 = Sum_{n=-oo..+oo} (-x)^n * (1 + A(x)*x^n)^n / A(x)^n.

(3) x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (x^n + A(x))^n.

(4) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (x^n + A(x))^n.

(5) 0 = Sum_{n=-oo..+oo} (-x)^n * (1 + A(x)^(n+1)*x^n)^n.

Example

G.f.: A(x) = 1 + x + 3*x^2 + 8*x^3 + 26*x^4 + 69*x^5 + 186*x^6 + 554*x^7 + 1606*x^8 + 4772*x^9 + 14825*x^10 + 45920*x^11 + 143586*x^12 + ...
where
x = ... - x^(-3)/(1 + A(x)*x^(-3))^3 + x^(-2)/(1 + A(x)*x^(-2))^2 - x^(-1)/(1 + A(x)*x^(-1)) + 1 - x*(1 + A(x)*x) + x^2*(1 + A(x)*x^2)^2 - x^3*(1 + A(x)*x^3)^3 + ... + (-x)^n*(1 + A(x)*x^n)^n + ...

Prog

(PARI) {a(n) = my(A=[1, 1]); for(i=1, n, A=concat(A, 0);
A[#A] = -polcoeff( sum(m=-#A, #A, (-x)^m*(1 + Ser(A)*x^m)^m ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1, 1]); for(i=1, n, A=concat(A, 0);
A[#A] = -polcoeff( sum(m=-#A, #A, (-1)^m*x^(m*(m-1))/(x^m + Ser(A))^m ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Sequence in context: A009452 A206141 A281972 * A343553 A317852 A306485

Adjacent sequences: A355866 A355867 A355868 * A355870 A355871 A355872

Keyword

nonn

Author

Paul D. Hanna, Aug 09 2022

Status

approved