A385444 Expansion of e.g.f. (1/x) * Series_Reversion( x/(4*x + sqrt(16*x^2+1))^(1/4) ).
1, 1, 3, 0, -195, -2160, 21735, 1290240, 13253625, -758419200, -34777667925, 0, 59136015863925, 2148944878080000, -60019159896320625, -8741374232887296000, -200253365886518319375, 23678097149478739968000, 2107410008390562322321875, 0, -11628675802354427876266081875
Offset: 0
Keywords
Programs
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PARI
a(n) = 8^n*n!*binomial((5*n+1)/8, n)/(5*n+1);
Formula
E.g.f.: (1/x) * Series_Reversion( x * exp(-arcsinh(4*x)/4) ).
E.g.f.: ( (1/x) * Series_Reversion( x/(1 + 8*x)^(5/8) ) )^(1/5).
E.g.f. A(x) satisfies A(x) = exp( (1/4) * arcsinh(4*x*A(x)) ).
E.g.f. A(x) satisfies A(x) = (1 + 8*x*A(x)^5)^(1/8).
a(n) = 8^n * n! * binomial((5*n+1)/8,n)/(5*n+1).
a(n) = Sum_{k=0..n} (n+1)^(k-1) * (4*i)^(n-k) * A385343(n,k), where i is the imaginary unit.
a(8*n+3) = 0 for n >= 0.