OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
a(n) ~ c * n! * n^(4*LambertW(1) - 1 + 2/(1 + LambertW(1))) / LambertW(1)^n, where c = 0.02048373460253911846... - Vaclav Kotesovec, Mar 13 2023
EXAMPLE
G.f.: A(x) = 1 + x + 4*x^2 + 29*x^3 + 294*x^4 + 3727*x^5 + 55748*x^6 + 950898*x^7 + 18094313*x^8 + 378363501*x^9 + 8600306451*x^10 + ...
The table of coefficients in the successive powers of g.f. A(x) begins:
n = 1: [1, 1, 4, 29, 294, 3727, 55748, ...];
n = 2: [1, 2, 9, 66, 662, 8274, 122143, ...];
n = 3: [1, 3, 15, 112, 1116, 13776, 200827, ...];
n = 4: [1, 4, 22, 168, 1669, 20384, 293654, ...];
n = 5: [1, 5, 30, 235, 2335, 28266, 402710, ...];
n = 6: [1, 6, 39, 314, 3129, 37608, 530334, ...];
n = 7: [1, 7, 49, 406, 4067, 48615, 679140, ...];
...
The table of coefficients in A(x)/(1 + x*A(x)^(n+2)) begins:
n = 1: [1, 0, 1, 13, 166, 2391, 38776, 699060, ...];
n = 2: [1, 0, 0, 7, 119, 1911, 32823, 612983, ...];
n = 3: [1, 0, -1, 0, 64, 1358, 26039, 515774, ...];
n = 4: [1, 0, -2, -8, 0, 724, 18356, 406634, ...];
n = 5: [1, 0, -3, -17, -74, 0, 9702, 284785, ...];
n = 6: [1, 0, -4, -27, -159, -824, 0, 149478, ...];
n = 7: [1, 0, -5, -38, -256, -1759, -10833, 0, ...];
...
in which the diagonal of all zeros illustrates that
[x^n] A(x) / (1 + x*A(x)^(n+2)) = 0 for n > 0.
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = -polcoeff( Ser(A)/(1 + x*Ser(A)^(#A+1)), #A-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 12 2023
STATUS
approved