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A349650 E.g.f. satisfies: A(x)^(A(x)^2) = 1 + x. 6
1, 1, -4, 36, -532, 11040, -295188, 9655772, -373422320, 16666348464, -843095987520, 47669276120928, -2979044176833888, 203906085094788960, -15170476121142482112, 1218972837861962011200, -105202043767190506428672, 9705514148732971389369600 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Eric Weisstein's World of Mathematics, Lambert W-Function.
FORMULA
a(n) = (-1)^(n-1) * Sum_{k=0..n} (2*k-1)^(k-1) * |Stirling1(n,k)|.
E.g.f. A(x) = -Sum_{k>=0} (2*k-1)^(k-1) * (-log(1+x))^k / k!.
E.g.f.: A(x) = ( 2*log(1+x)/LambertW(2*log(1+x)) )^(1/2).
a(n) ~ -(-1)^n * n^(n-1) * exp(n*(exp(-1)/2 - 1)) / (sqrt(2) * (exp(exp(-1)/2) - 1)^(n - 1/2)). - Vaclav Kotesovec, Nov 24 2021
MATHEMATICA
nmax = 20; A[_] = 1;
Do[A[x_] = (1 + x)^(1/A[x]^2) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
PROG
(PARI) a(n) = (-1)^(n-1)*sum(k=0, n, (2*k-1)^(k-1)*abs(stirling(n, k, 1)));
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(-sum(k=0, N, (2*k-1)^(k-1)*(-log(1+x))^k/k!)))
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((2*log(1+x)/lambertw(2*log(1+x)))^(1/2)))
CROSSREFS
Sequence in context: A179422 A098629 A308333 * A294359 A336639 A178184
KEYWORD
sign
AUTHOR
Seiichi Manyama, Nov 23 2021
STATUS
approved

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Last modified June 25 01:38 EDT 2024. Contains 373691 sequences. (Running on oeis4.)