A300281 G.f.: Sum_{n>=0} (1 + (1 + 2*x)^n)^n / 2^(2*n+1).

1, 3, 41, 967, 32109, 1373603, 71889889, 4448939407, 317785091933, 25731184562939, 2328915407063705, 233001395274021991, 25533207271295208821, 3041514682215417132371, 391307447238067445930129, 54074977650384006192679103, 7988238906084714854241917421, 1256227929202274469473017312811, 209531162751200464078327250379657, 36946198191974438054673167074349079

Offset

0,2

Comments

Is there a closed-form expression for the terms of this sequence?

Links

Table of n, a(n) for n=0..19.

Formula

G.f. A(x) is given by:

(1) A(x) = Sum_{n>=0} 2 * (1 + (1 + 2*x)^n)^n / 4^(n+1).

(2) A(x) = Sum_{n>=0} 2 * (1 + 2*x)^(n^2) / (4 - (1 + 2*x)^n)^(n+1).

Example

G.f.: A(x) = 1 + 3*x + 41*x^2 + 967*x^3 + 32109*x^4 + 1373603*x^5 + 71889889*x^6 + 4448939407*x^7 + 317785091933*x^8 + 25731184562939*x^9 + ...
such that
A(x) = 1/2 + (1 + (1+2*x))/2^3 + (1 + (1+2*x)^2)^2/2^5 + (1 + (1+2*x)^3)^3/2^7 + (1 + (1+2*x)^4)^4/2^9 + (1 + (1+2*x)^5)^5/2^11 + (1 + (1+2*x)^6)^6/2^13 + ...
Also, due to a series identity,
A(x) = 2/3 + 2*(1+2*x)/(4 - (1+2*x))^2 + 2*(1+2*x)^4/(4 - (1+2*x)^2)^3 + 2*(1+2*x)^9/(4 - (1+2*x)^3)^4 + 2*(1+2*x)^16/(4 - (1+2*x)^4)^5 + 2*(1+2*x)^25/(4 - (1+2*x)^5)^6 + 2*(1+2*x)^36/(4 - (1+2*x)^6)^7 + ...

Crossrefs

Sequence in context: A138961 A075022 A367981 * A012175 A007313 A222524

Adjacent sequences: A300278 A300279 A300280 * A300282 A300283 A300284

Keyword

nonn

Author

Paul D. Hanna, Mar 03 2018

Status

approved