A220231 G.f.: Sum_{n>=1} (2 - (1-x)^n) * (1 - (1-x)^n)^(n-1) / (1-x)^(n^2).

1, 4, 22, 184, 2094, 30300, 532918, 11041936, 263451958, 7114155240, 214501732236, 7143254501280, 260397686968140, 10313589262630724, 441025376697310226, 20250545724670230240, 993756093394720170206, 51903607891071021948092, 2874779134773855301743682

Offset

0,2

Comments

Compare the g.f. of this sequence to the identity (when G(x) = 1/(1-x)):

1 = Sum_{n>=1} (2*G(x)^n - 1) * (1 - G(x)^n)^(n-1) for all G(x) such that G(0)=1.

Links

Table of n, a(n) for n=0..18.

Formula

G.f.: 1 + Sum_{n>=1} 2*(2 - (1-x)^(2*n)) * (1 - (1-x)^(2*n))^(2*n-1) / (1-x)^(4*n^2).

G.f.: -1 + Sum_{n>=0} 2*(2 - (1-x)^(2*n+1))*(1 - (1-x)^(2*n+1))^(2*n)/(1-x)^((2*n+1)^2).

Example

G.f.: A(x) = 1 + 4*x + 22*x^2 + 184*x^3 + 2094*x^4 + 30300*x^5 +...
where
A(x) = (1+x)/(1-x) + (1+2*x-x^2)*(2*x-x^2)/(1-x)^4 + (1+3*x-3*x^2+x^3)*(3*x-3*x^2+x^3)^2/(1-x)^9 + (1+4*x-6*x^2+4*x^3-x^4)*(4*x-6*x^2+4*x^3-x^4)^3/(1-x)^16 +...
Compare the g.f. to the identity:
1 = (1+x)/(1-x) - (1+2*x-x^2)*(2*x-x^2)/(1-x)^4 + (1+3*x-3*x^2+x^3)*(3*x-3*x^2+x^3)^2/(1-x)^9 - (1+4*x-6*x^2+4*x^3-x^4)*(4*x-6*x^2+4*x^3-x^4)^3/(1-x)^16 +-...

Prog

(PARI) {a(n)=polcoeff(sum(m=1, n+1, (2 - (1-x)^m) * (1 - (1-x)^m +x*O(x^n))^(m-1)/(1-x+x*O(x^n))^(m^2)), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=polcoeff(1+sum(m=1, n\2+1, 2*(2 - (1-x)^(2*m)) * (1 - (1-x)^(2*m) +x*O(x^n))^(2*m-1)/(1-x+x*O(x^n))^(4*m^2)), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=polcoeff(-1+sum(m=0, n\2, 2*(2 - (1-x)^(2*m+1)) * (1 - (1-x)^(2*m+1) +x*O(x^n))^(2*m)/(1-x+x*O(x^n))^((2*m+1)^2)), n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A220266.

Sequence in context: A362747 A222885 A235329 * A112370 A197961 A203120

Adjacent sequences: A220228 A220229 A220230 * A220232 A220233 A220234

Keyword

nonn

Author

Paul D. Hanna, Dec 11 2012

Status

approved