login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A220266 G.f.: Sum_{n>=1} (2*(1+x)^n - 1) * ((1+x)^n - 1)^(n-1). 3
1, 4, 18, 144, 1604, 22944, 400624, 8259680, 196358760, 5287879092, 159094582274, 5288950560768, 192527721428892, 7616404083126180, 325361411700398046, 14926683772801407168, 731947910056020737036, 38204289826040411251632, 2114787166947079113869760 (list; graph; refs; listen; history; text; internal format)
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: A304997 A060841 A059837 * A218917 A054759 A286630

Adjacent sequences:  A220263 A220264 A220265 * A220267 A220268 A220269

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Dec 09 2012

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 6 15:57 EST 2021. Contains 349565 sequences. (Running on oeis4.)