A326276 G.f.: Sum_{n>=0} (1 + (1+x)^(n+1))^n * x^n.

1, 2, 6, 21, 85, 382, 1879, 9986, 56818, 343640, 2196596, 14770122, 104063085, 765661874, 5866191429, 46683934520, 385048724001, 3285146877603, 28942067342876, 262882422213165, 2458316711782337, 23637510378534754, 233423898596027454, 2364847720082290621, 24555411743247510317, 261085211212909391915

Offset: 0

Author

Paul D. Hanna, Jun 28 2019

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 21*x^3 + 85*x^4 + 382*x^5 + 1879*x^6 + 9986*x^7 + 56818*x^8 + 343640*x^9 + 2196596*x^10 + ...
such that
A(x) = 1 + (1 + (1+x)^2)*x + (1 + (1+x)^3)^2*x^2 + (1 + (1+x)^4)^3*x^3 + (1 + (1+x)^5)^4*x^4 + ... + (1 + (1+x)^(n+1))^n*x^n + ...
also
A(x) = 1/(1 - x) + (1+x)^2*x/(1 - x*(1+x))^2 + (1+x)^6*x^2/(1 - x*(1+x)^2)^3 + (1+x)^12*x^3/(1 - x*(1+x)^3)^4 + ... + (1+x)^(n*(n+1))*x^n/(1 - x*(1+x)^n)^(n+1) + ...
RELATED SERIES.
Below we illustrate the following identity at specific values of x:
Sum_{n>=0} (1 + (1+x)^(n+1))^n * x^n  =  Sum_{n>=0} (1+x)^(n*(n+1)) * x^n / (1 - x*(1+x)^n)^(n+1).
(1) At x = -1/2, the following sums are equal
S1 = Sum_{n>=0} (-1)^n * (2^(n+1) + 1)^n / 2^(n*(n+2)),
S1 = Sum_{n>=0} (-1)^n * 2 / (2^(n+1) + 1)^(n+1),
where S1 = 0.58938625589631021783349702645576048800172938765646329470992...
(2) At x = -1/3, the following sums are equal
S2 = Sum_{n>=0} (-1)^n * (2^(n+1) + 3^(n+1))^n / 3^(n*(n+2)),
S2 = Sum_{n>=0} (-1)^n * 3 * 2^(n*(n+1)) / (3^(n+1) + 2^n)^(n+1),
where S2 = 0.65707817941052544107009145640756914928885409483935267126701...
(3) At x = -2/3, the following sums are equal
S3 = Sum_{n>=0} (-2)^n * (3^(n+1) + 1)^n / 3^(n*(n+2)),
S3 = Sum_{n>=0} (-2)^n * 3 / (3^(n+1) + 2)^(n+1),
where S3 = 0.55090474258125970373130850821926676214280685554645756713729...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A301306.

Programs

PARI

{a(n) = polcoeff( sum(m=0,n, (1 + (1+x)^(m+1) +x*O(x^n) )^m * x^m), n)}
for(n=0,30,print1(a(n),", "))

Formula

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

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