A385501 -id:A385501 - cp's OEIS frontend

A385500 Expansion of e.g.f. exp( -LambertW(-arctanh(x)) ).

1, 1, 3, 18, 149, 1640, 22359, 366128, 6998697, 153191808, 3779353515, 103800229632, 3141633970749, 103904351855616, 3728602377979647, 144297781732300800, 5991021498320041809, 265636734347975688192, 12527923794824003280723, 626224876080360687599616

Offset: 0

Seiichi Manyama, Jul 01 2025

nonn

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A385501, A385502.

Cf. A111594, A385425.

nmax=19; CoefficientList[Series[Exp[ -LambertW[-ArcTanh[x]]],{x,0,nmax}],x]Range[0,nmax]! (* Stefano Spezia, Jul 01 2025 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-atanh(x)))))

E.g.f. A(x) satisfies A(x) = exp( arctanh(x) * A(x) ).
E.g.f. A(x) satisfies A(x) = ( (1+x)/(1-x) )^(A(x)/2).
a(n) = Sum_{k=0..n} (k+1)^(k-1) * A111594(n,k).
a(n) ~ sqrt(exp(4*exp(-1)) - 1) * n^(n-1) / (2*exp(n - 3/2 + exp(-1)) * tanh(exp(-1))^n). - Vaclav Kotesovec, Jul 01 2025

A385502 E.g.f. A(x) satisfies A(x) = exp( arctanh(x * A(x)) / A(x) ).

Original entry on oeis.org

1, 1, 1, 3, 25, 205, 2001, 25991, 394353, 6718041, 130319745, 2833146987, 67767170505, 1772434086501, 50392083769041, 1546052750636655, 50905035315373281, 1790951445870568113, 67050161599822764417, 2661363261252799648083, 111637709182606749500025

Offset: 0

Seiichi Manyama, Jul 01 2025

nonn

Cf. A385500, A385501.

Cf. A111594, A385428.

terms = 21; A[] = 1; Do[A[x] = Exp[ArcTanh[x*A[x]]/A[x]] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x]Range[0,terms-1]! (* Stefano Spezia, Jul 01 2025 *)

a111594(n, k) = my(x='x+O('x^(n+1))); n!*polcoef(atanh(x)^k/k!, n);
a(n) = sum(k=0, n, (n-k+1)^(k-1)*a111594(n, k));

E.g.f. A(x) satisfies A(x) = ( (1+x*A(x))/(1-x*A(x)) )^(1/(2*A(x))).
a(n) = Sum_{k=0..n} (n-k+1)^(k-1) * A111594(n,k).
a(n) ~ s * sqrt((-1 + r^2*s^2)/(1 - 2*r^2*(1 + r)*s^2 + r^4*s^4)) * n^(n-1) / (exp(n) * r^(n - 1/2)), where r = 0.4422573061236400123439455590007131605377062990202... and s = 1.93686591146053883124948614770176661449449740697... are the roots of the system of equations ((1 + r*s)/(1 - r*s))^(1/2/s) = s, 2*r*s + (-1 + r^2*s^2)*(log((1 + r*s)/(1 - r*s)) + 2*s) = 0. - Vaclav Kotesovec, Jul 01 2025

cp's OEIS Frontend

A385500 Expansion of e.g.f. exp( -LambertW(-arctanh(x)) ).

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