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%I #8 Aug 09 2018 09:43:23
%S 1,4,70,2180,95729,5422192,375951144,30833206304,2919367902648,
%T 313380517364324,37606931999739230,4988933437333555060,
%U 724960700435104219679,114519163835687116024256,19538926882901715534673728,3580844611314789257667535968,701546780854024941112271649610,146318317830136401429653726419700,32367591848747955557013839920695374,7569528177000020896435962191564396740
%N G.f.: Sum_{n>=0} ((1+x)^(4*n) - 1)^n.
%H Vaclav Kotesovec, <a href="/A301586/b301586.txt">Table of n, a(n) for n = 0..314</a>
%F G.f.: Sum_{n>=0} (1+x)^(4*n^2) /(1 + (1+x)^(4*n))^(n+1).
%F a(n) ~ c * d^n * n! / sqrt(n), where d = 4*A317855 = 12.64435461546171525532068881035252996690553109675422536650911283015078823687... and c = 0.31492557816516652573983016205911709623053... - _Vaclav Kotesovec_, Aug 09 2018
%e G.f.: A(x) = 1 + 4*x + 70*x^2 + 2180*x^3 + 95729*x^4 + 5422192*x^5 + 375951144*x^6 + 30833206304*x^7 + ...
%e such that
%e A(x) = 1 + ((1+x)^4-1) + ((1+x)^8-1)^2 + ((1+x)^12-1)^3 + ((1+x)^16-1)^4 + ((1+x)^20-1)^5 + ((1+x)^24-1)^6 + ((1+x)^28-1)^7 + ...
%e Also,
%e A(x) = 1/2 + (1+x)^4/(1 + (1+x)^4)^2 + (1+x)^16/(1 + (1+x)^8)^3 + (1+x)^36/(1 + (1+x)^12)^4 + (1+x)^64/(1 + (1+x)^16)^5 + (1+x)^100/(1 + (1+x)^20)^6 + ...
%o (PARI) {a(n) = my(A,o=x*O(x^n)); A = sum(m=0,n, ((1+x +o)^(4*m) - 1)^m ); polcoeff(A,n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A122400, A301584, A301585.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Mar 24 2018
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