A301586 G.f.: Sum_{n>=0} ((1+x)^(4*n) - 1)^n.

1, 4, 70, 2180, 95729, 5422192, 375951144, 30833206304, 2919367902648, 313380517364324, 37606931999739230, 4988933437333555060, 724960700435104219679, 114519163835687116024256, 19538926882901715534673728, 3580844611314789257667535968, 701546780854024941112271649610, 146318317830136401429653726419700, 32367591848747955557013839920695374, 7569528177000020896435962191564396740

Offset: 0

Author

Paul D. Hanna, Mar 24 2018

Keywords

• nonn

Examples

			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 + ...
such that
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 + ...
Also,
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 + ...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..314

Crossrefs

Cf. A122400, A301584, A301585.

Programs

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)}
for(n=0,30,print1(a(n),", "))

Formula

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

a(n) ~ c * d^n * n! / sqrt(n), where d = 4*A317855 = 12.64435461546171525532068881035252996690553109675422536650911283015078823687... and c = 0.31492557816516652573983016205911709623053... - Vaclav Kotesovec, Aug 09 2018