A372235 E.g.f. A(x) satisfies A(x) = exp( x * (1 + A(x)^(3/2)) ).
1, 2, 10, 98, 1456, 29132, 734932, 22407464, 801710560, 32940601424, 1528816004944, 79109107128944, 4516145972879680, 281970941337424640, 19114791434098402816, 1398205517746364523008, 109771912847021666795008, 9206931548976575570314496
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Keywords
Links
- Eric Weisstein's World of Mathematics, Lambert W-Function.
Programs
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PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-2/3*lambertw(-3*x/2*exp(3*x/2))))) -
PARI
a(n, r=1, t=0, u=3/2) = r*sum(k=0, n, (t*n+u*k+r)^(n-1)*binomial(n, k));
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PARI
my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (3*k/2+1)^(k-1)*x^k/(1-(3*k/2+1)*x)^(k+1)))
Formula
E.g.f.: A(x) = exp( x - 2/3 * LambertW(-3*x/2 * exp(3*x/2)) ).
If e.g.f. satisfies A(x) = exp( r*x*A(x)^(t/r) * (1 + A(x)^(u/r)) ), then a(n) = r * Sum_{k=0..n} (t*n+u*k+r)^(n-1) * binomial(n,k).
G.f.: Sum_{k>=0} (3*k/2+1)^(k-1) * x^k/(1 - (3*k/2+1)*x)^(k+1).
a(n) ~ sqrt(1 + LambertW(exp(-1))) * 3^(n-1) * n^(n-1) / (2^(n-1) * exp(n) * LambertW(exp(-1))^(n + 2/3)). - Vaclav Kotesovec, Apr 24 2024