A302615 G.f.: Sum_{n>=0} (3 + (1+x)^n)^n / (4 + (1+x)^n)^(n+1).

1, 1, 10, 130, 2390, 56714, 1644138, 56327820, 2226708772, 99761490536, 4995375316146, 276464859358474, 16757956600528786, 1104116777798713154, 78565751676021256606, 6004629888868350015506, 490572645247461234631946, 42665124626946741636482996, 3935474733572880332326074450, 383756013888633346483785849474

Offset: 0

Author

Paul D. Hanna, Apr 10 2018

Keywords

• nonn

Comments

The following identity holds for |y| <= 1 and fixed real k > 0:

Sum_{n>=0} (k + y^n)^n/(1+k + y^n)^(n+1) = Sum_{n>=0} (y^n - 1)^n/(1+k - k*y^n)^(n+1).

Examples

			G.f.: A(x) = 1 + x + 10*x^2 + 130*x^3 + 2390*x^4 + 56714*x^5 + 1644138*x^6 + 56327820*x^7 + 2226708772*x^8 + 99761490536*x^9 + ...
such that
A(x) = 1/5  +  (3 + (1+x))/(4 + (1+x))^2  +  (3 + (1+x)^2)^2/(4 + (1+x)^2)^3  +  (3 + (1+x)^3)^3/(4 + (1+x)^3)^4  +  (3 + (1+x)^4)^4/(4 + (1+x)^4)^5  +  (3 + (1+x)^5)^5/(4 + (1+x)^5)^6  +  (3 + (1+x)^6)^6/(4 + (1+x)^6)^7  + ...
Also,
A(x) = 1  +  ((1+x) - 1)/(4 - 3*(1+x))^2  +  ((1+x)^2 - 1)^2/(4 - 3*(1+x)^2)^3  +  ((1+x)^3 - 1)^3/(4 - 3*(1+x)^3)^4  +  ((1+x)^4 - 1)^4/(4 - 3*(1+x)^4)^5  +  ((1+x)^5 - 1)^5/(4 - 3*(1+x)^5)^6  +  ((1+x)^6 - 1)^6/(4 - 3*(1+x)^6)^7 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A122400, A302598, A302614.

Programs

PARI

{a(n) = my(A=1, o=x*O(x^n)); A = sum(m=0, n, ((1+x +o)^m - 1)^m / (4 - 3*(1+x +o)^m)^(m+1)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f.: Sum_{n>=0} ((1+x)^n - 1)^n / (4 - 3*(1+x)^n)^(n+1).

a(n) ~ c * d^n * n! / sqrt(n), where d = 5.2709551504518355656831902094014170087... and c = 0.26621450180820822374221893929... - Vaclav Kotesovec, Aug 09 2018