login
A301463
G.f.: Sum_{n>=0} (2*(1+x)^n - 1)^n / 2^(n+1).
6
1, 6, 134, 5102, 272694, 18758134, 1577807110, 156883546142, 18001728695894, 2341268080847014, 340346951612008454, 54686371000455538574, 9624103747115691611318, 1841049154379441320293142, 380367456989975381891133446, 84407842226680664984458744126, 20023121531700221583865582432854, 5056357801144690975957652265658438, 1354259474931265421064754160458035078, 383444904170987865090156939638756172846
OFFSET
0,2
LINKS
FORMULA
G.f.: Sum_{n>=0} 2^n * (1+x)^(n^2) / (2 + (1+x)^n)^(n+1).
G.f.: Sum_{n>=0} ((1+x)^n - 1/2)^n / 2.
a(n) ~ c * d^n * n! / sqrt(n), where d = 15.305828173910545025228605110120647795... and c = 0.4246982835243422293505427496472772728... - Vaclav Kotesovec, Aug 09 2018
EXAMPLE
G.f.: A(x) = 1 + 6*x + 134*x^2 + 5102*x^3 + 272694*x^4 + 18758134*x^5 + 1577807110*x^6 + 156883546142*x^7 + 18001728695894*x^8 + ...
such that
A(x) = 1/2 + (2*(1+x) - 1)/2^2 + (2*(1+x)^2 - 1)^2/2^3 + (2*(1+x)^3 - 1)^3/2^4 + (2*(1+x)^4 - 1)^4/2^5 + (2*(1+x)^5 - 1)^5/2^6 + ...
Also,
A(x) = 1/3 + 2*(1+x)/(2 + (1+x))^2 + 2^2*(1+x)^4/(2 + (1+x)^2)^3 + 2^3*(1+x)^9/(2 + (1+x)^3)^4 + 2^4*(1+x)^16/(2 + (1+x)^4)^5 + 2^5*(1+x)^25/(2 + (1+x)^5)^6 + 2^6*(1+x)^36/(2 + (1+x)^6)^7 + ...
MATHEMATICA
nmax = 20; Round[CoefficientList[Series[Sum[(2*(1 + x)^j - 1)^j/2^(j + 1), {j, 0, nmax^2}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Oct 08 2020 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 24 2018
EXTENSIONS
b-file confirmed by Vaclav Kotesovec, Oct 08 2020
STATUS
approved