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A301583 G.f.: Sum_{n>=0} 4^n * ((1+x)^n - 1)^n. 7

%I #16 Oct 13 2020 03:27:02

%S 1,4,64,1792,70736,3600128,224255040,16521605376,1405131880000,

%T 135480346104896,14602769310474240,1739917222954854400,

%U 227081534040721917952,32217108743091290851328,4936803887495636263284736,812576030237749532251019264,142976863303365903802301729024,26781577193841845859144244087808,5320767287406003709062843236972544,1117525692987087894816123931091214336

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

%C In general, if k > 0 and g.f.: Sum_{j>=0} k^j * ((1+x)^j - 1)^j, then a(n) ~ c * (1 + k*exp(1/r))^n * r^(2*n) * n! / sqrt(n), where r is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/k and c is a constant (dependent only on k). - _Vaclav Kotesovec_, Oct 08 2020

%H Vaclav Kotesovec, <a href="/A301583/b301583.txt">Table of n, a(n) for n = 0..318</a>

%F G.f.: Sum_{n>=0} 4^n * (1+x)^(n^2) /(1 + 4*(1+x)^n)^(n+1).

%F a(n) ~ c * d^n * n! / sqrt(n), where d = (1 + 4*exp(1/r)) * r^2 = 11.35554580636894436474777793373210745006910386794268638744346793426715754570218..., where r = 0.95894043087329419322124137165060249611787608513866855417024... is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/4 and c = 0.358692703763731594549618907599728117285634153... - _Vaclav Kotesovec_, Aug 09 2018, updated Oct 08 2020

%e G.f.: A(x) = 1 + 4*x + 64*x^2 + 1792*x^3 + 70736*x^4 + 3600128*x^5 + 224255040*x^6 + 16521605376*x^7 + 1405131880000*x^8 + ...

%e such that

%e A(x) = 1 + 4*((1+x)-1) + 16*((1+x)^2-1)^2 + 64*((1+x)^3-1)^3 + 256*((1+x)^4-1)^4 + 1024*((1+x)^5-1)^5 + 4096*((1+x)^6-1)^6 + ...

%e Also,

%e A(x) = 1/5 + 4*(1+x)/(1 + 4*(1+x))^2 + 16*(1+x)^4/(1 + 4*(1+x)^2)^3 + 64*(1+x)^9/(1 + 4*(1+x)^3)^4 + 256*(1+x)^16/(1 + 4*(1+x)^4)^5 + 1024*(1+x)^25/(1 + 4*(1+x)^5)^6 + ...

%t nmax = 20; CoefficientList[Series[1 + Sum[4^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, 4^m * ((1+x +o)^m - 1)^m ); polcoeff(A,n)}

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

%Y Cf. A122400, A301581, A301582, A301463.

%Y Cf. A122399, A195005, A195263, A338040.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Mar 24 2018

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