A381141 Expansion of e.g.f. exp( -LambertW(-x * cos(x)) ).

1, 1, 3, 13, 89, 821, 9667, 137817, 2306705, 44308009, 960645251, 23205700453, 618086944873, 17996847978461, 568729575572355, 19387150575025201, 709130794848586657, 27704208465508996945, 1151379111946617111043, 50721472225191792506301, 2360928161776701549045241

Offset

0,3

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Links

Table of n, a(n) for n=0..20.

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

Formula

E.g.f. A(x) satisfies A(x) = exp( x * cos(x) * A(x) ).

a(n) = Sum_{k=0..n} (k+1)^(k-1) * i^(n-k) * A185951(n,k), where i is the imaginary unit.

a(n) ~ sqrt(1 - r*sqrt(exp(2)*r^2 - 1)) * n^(n-1) / (exp(n-1) * r^n), where r = 0.3992882914671621302380071869880891621973411296719... is the root of the equation r*cos(r) = exp(-1). - Vaclav Kotesovec, Jan 30 2026

Prog

(PARI) a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, (k+1)^(k-1)*I^(n-k)*a185951(n, k));

Crossrefs

Cf. A009189, A381144, A381146.

Cf. A185951.

Sequence in context: A373658 A097711 A114477 * A345104 A116434 A174290

Adjacent sequences: A381138 A381139 A381140 * A381142 A381143 A381144

Keyword

nonn

Author

Seiichi Manyama, Feb 15 2025

Status

approved