A303062 G.f. A(x) satisfies: [x^(n-1)] (1 + x*A(x)^(n-1))^n / A(x)^n = 0 for n>1.

1, 1, 2, 8, 60, 643, 8564, 133890, 2376261, 46832442, 1009739331, 23564025488, 590503218735, 15793704933899, 448695132962248, 13487808514722460, 427624923581550100, 14260707306806609885, 499071020445057149835, 18290961984686434723480, 700757535935308305865473, 28017787701624063252219677

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

a(n) ~ c * n! * n^(2*LambertW(1)) / LambertW(1)^n, where c = 0.153879081661359639962985708... - Vaclav Kotesovec, Aug 11 2021

Example

G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 60*x^4 + 643*x^5 + 8564*x^6 + 133890*x^7 + 2376261*x^8 + 46832442*x^9 + 1009739331*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients in (1 + x*A(x)^(n-1))^n / A(x)^n begins:
n=1: [1, 0, -2, -6, -50, -565, -7731, -122983, ...];
n=2: [1, 0, -2, -10, -89, -1030, -14307, -230054, ...];
n=3: [1, 0, 0, -9, -111, -1380, -19677, -320958, ...];
n=4: [1, 0, 4, 0, -94, -1520, -23388, -392776, ...];
n=5: [1, 0, 10, 20, 0, -1210, -24030, -436250, ...];
n=6: [1, 0, 18, 54, 225, 0, -18345, -427944, ...];
n=7: [1, 0, 28, 105, 651, 2835, 0, -316344, ...];
n=8: [1, 0, 40, 176, 1364, 8360, 41976, 0, ...]; ...
in which the main diagonal equals all zeros after the initial term, illustrating that [x^(n-1)] (1 + x*A(x)^(n-1))^n / A(x)^n = 0 for n>1.

Prog

(PARI) {a(n) = my(A=[1]); for(m=1, n+1, A=concat(A, 0); A[m] = Vec( (1 + x*Ser(A)^(m-1))^m/Ser(A)^m )[m]/m ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A303063, A302702.

Sequence in context: A208124 A052622 A303672 * A001188 A355100 A303532

Adjacent sequences: A303059 A303060 A303061 * A303063 A303064 A303065

Keyword

nonn

Author

Paul D. Hanna, Apr 17 2018

Status

approved