A332048 a(n) = n! * [x^n] 1 / (1 - LambertW(x))^n.

1, 1, 2, 15, 104, 1145, 13824, 208831, 3536000, 68918769, 1489702400, 35742514511, 937323767808, 26750313223465, 824073079660544, 27276657371589375, 965004380380626944, 36347144974616190689, 1451974448007830568960, 61319892272079181137679, 2729671240750270054400000

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..394

Formula

a(n) = Sum_{k=0..n} Sum_{j=0..n-1} (-1)^(n - k) * binomial(n - 1, j) * Stirling1(j + 1, k) * n^(n + k - j - 1) for n > 0.

a(n) ~ phi^(3*n + 1/2) * n^n / (5^(1/4) * exp(n + n/phi)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Feb 07 2020

Mathematica

Table[n! SeriesCoefficient[1/(1 - LambertW[x])^n, {x, 0, n}], {n, 0, 20}]
Join[{1}, Table[Sum[Sum[(-1)^(n - k) Binomial[n - 1, j] StirlingS1[j + 1, k] n^(n + k - j - 1), {j, 0, n - 1}], {k, 0, n}], {n, 1, 20}]]

Crossrefs

Cf. A239761, A277458, A277510.

Sequence in context: A081018 A006675 A215643 * A295268 A037524 A037733

Adjacent sequences: A332045 A332046 A332047 * A332049 A332050 A332051

Keyword

nonn

Author

Ilya Gutkovskiy, Feb 06 2020

Status

approved