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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..100

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

Cf. A122400, A301581, A301582, A301583.

Sequence in context: A179564 A263583 A295408 * A220064 A050281 A096756

Adjacent sequences:  A301460 A301461 A301462 * A301464 A301465 A301466

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Mar 24 2018

EXTENSIONS

b-file confirmed by Vaclav Kotesovec, Oct 08 2020

STATUS

approved

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Last modified December 2 01:01 EST 2020. Contains 338864 sequences. (Running on oeis4.)