A220266 G.f.: Sum_{n>=1} (2*(1+x)^n - 1) * ((1+x)^n - 1)^(n-1).

1, 4, 18, 144, 1604, 22944, 400624, 8259680, 196358760, 5287879092, 159094582274, 5288950560768, 192527721428892, 7616404083126180, 325361411700398046, 14926683772801407168, 731947910056020737036, 38204289826040411251632, 2114787166947079113869760

Offset

0,2

Comments

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

Equals the antidiagonal sums of triangle A220265:

a(n) = Sum_{k=0..n} A220265(n-k+1,k) for n>=0.

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

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

Example

G.f.: A(x) = 1 + 4*x + 18*x^2 + 144*x^3 + 1604*x^4 + 22944*x^5 +...
where
A(x) = (1+2*x) + (1+4*x+2*x^2)*(2*x+x^2) + (1+6*x+6*x^2+2*x^3)*(3*x+3*x^2+x^3)^2 + (1+8*x+12*x^2+8*x^3+2*x^4)*(4*x+6*x^2+4*x^3+x^4)^3 +...
Compare the g.f. to the identity:
1 = (1+2*x) - (1+4*x+2*x^2)*(2*x+x^2) + (1+6*x+6*x^2+2*x^3)*(3*x+3*x^2+x^3)^2 - (1+8*x+12*x^2+8*x^3+2*x^4)*(4*x+6*x^2+4*x^3+x^4)^3 +-...

Prog

(PARI) {a(n)=polcoeff(sum(m=1, n+1, (2*(1+x)^m - 1) * ((1+x)^m - 1 +x*O(x^n))^(m-1)), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* As Row Sums of Triangle A220265: */
{A220265(n, k)=polcoeff((2*(1+x)^n-1)*((1+x)^n-1)^(n-1)/x^(n-1), k)}
{a(n)=sum(k=0, n, A220265(n-k+1, k))}
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) - 1 +x*O(x^n))^(2*m-1)), 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) - 1 +x*O(x^n))^(2*m)), n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A220265, A220231.

Sequence in context: A370774 A060841 A059837 * A218917 A054759 A286630

Adjacent sequences: A220263 A220264 A220265 * A220267 A220268 A220269

Keyword

nonn

Author

Paul D. Hanna, Dec 09 2012

Status

approved