A301582 - OEIS

%I #15 Oct 08 2020 04:24:17

%S 1,3,36,765,22932,886707,41971041,2349915543,151893243711,

%T 11131097539221,911906584505874,82586031357156975,8192750710914222984,

%U 883506535094875209327,102907862475072248379060,12875067336646598300376165,1722014444866824121524712497,245185575019136812676809863351,37027348593726417935247243009495,5911490521308027393188499233189367,994821814352463817234026392636083551

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

%H Vaclav Kotesovec, <a href="/A301582/b301582.txt">Table of n, a(n) for n = 0..329</a>

%F G.f.: Sum_{n>=0} 3^n * (1+x)^(n^2) /(1 + 3*(1+x)^n)^(n+1).

%F a(n) ~ c * d^n * n! / sqrt(n), where d = (1 + 3*exp(1/r)) * r^2 = 8.632012704198046828204904686098781240870113556702123911346365466059061495897353..., where r = 0.947093169766093813913446822751643203941993193936... is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/3 and c = 0.34734097623709084937300542950550592394946492732014... - _Vaclav Kotesovec_, Aug 09 2018

%e G.f.: A(x) = 1 + 3*x + 36*x^2 + 765*x^3 + 22932*x^4 + 886707*x^5 + 41971041*x^6 + 2349915543*x^7 + 151893243711*x^8 + ...

%e such that

%e A(x) = 1 + 3*((1+x)-1) + 9*((1+x)^2-1)^2 + 27*((1+x)^3-1)^3 + 81*((1+x)^4-1)^4 + 243*((1+x)^5-1)^5 + 729*((1+x)^6-1)^6 + 2187*((1+x)^7-1)^7 + ...

%e Also,

%e A(x) = 1/4 + 3*(1+x)/(1 + 3*(1+x))^2 + 9*(1+x)^4/(1 + 3*(1+x)^2)^3 + 27*(1+x)^9/(1 + 3*(1+x)^3)^4 + 81*(1+x)^16/(1 + 3*(1+x)^4)^5 + 243*(1+x)^25/(1 + 3*(1+x)^5)^6 + ...

%t nmax = 20; CoefficientList[Series[1 + Sum[3^j*((1 + x)^j - 1)^j, {j, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Oct 08 2020 *)

%o (PARI) {a(n) = my(A,o=x*O(x^n)); A = sum(m=0,n, 3^m * ((1+x +o)^m - 1)^m ); polcoeff(A,n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A122400, A195263, A301581, A301583, A301463.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Mar 24 2018