OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..400
FORMULA
a(n) ~ c * 2^(2*n) * n^(n - 3/2) / (exp(n) * w^n * (2-w)^n), where w = -LambertW(-2*exp(-2)) = -A226775 and c = 11.13179071... - Vaclav Kotesovec, Feb 25 2026
EXAMPLE
O.g.f.: A(x) = 1 + x + 8*x^2 + 122*x^3 + 2690*x^4 + 76599*x^5 + 2651334*x^6 + 107439122*x^7 + 4969179466*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n*(n+1)*x*A(x)^3) / A(x)^(n+1) for n >= 0 begins
n = 0: [1, -1, -14, -642, -57720, -8375280, ...];
n = 1: [1, 0, -18, -1048, -101208, -15193728, ...];
n = 2: [1, 3, 0, -1140, -136584, -21676464, ...];
n = 3: [1, 8, 76, 0, -141144, -27591168, ...];
n = 4: [1, 15, 270, 5210, 0, -27956520, ...];
n = 5: [1, 24, 666, 21048, 671040, 0, ...];
in which the main diagonal is all zeros after the initial term.
The above table is derived from the following series
n = 1: exp(1*2*x*A(x)^3)/A(x)^2 = 1 + 0*x - 18*x^2/2! - 1048*x^3/3! - 101208*x^4/4! - 15193728*x^5/5! + ...
n = 2: exp(2*3*x*A(x)^3)/A(x)^3 = 1 + 3*x + 0*x^2/2! - 1140*x^3/3! - 136584*x^4/4! - 21676464*x^5/5! + ...
n = 3: exp(3*4*x*A(x)^3)/A(x)^4 = 1 + 8*x + 76*x^2/2! + 0*x^3/3! - 141144*x^4/4! - 27591168*x^5/5! + ...
n = 4: exp(4*5*x*A(x)^3)/A(x)^5 = 1 + 15*x + 270*x^2/2! + 5210*x^3/3! + 0*x^4/4! - 27956520*x^5/5! + ...
n = 5: exp(5*6*x*A(x)^3)/A(x)^6 = 1 + 24*x + 666*x^2/2! + 21048*x^3/3! + 671040*x^4/4! + 0*x^5/5! + ...
...
illustrating that [x^n] exp(n*(n+1)*x*A(x)^3) / A(x)^(n+1) = 0 for n > 0.
PROG
(PARI) {a(n) = my(A=[1], m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp((m-1)*m*x*Ser(A)^3 +x*O(x^m)) / Ser(A)^m )[m]/m ); A[n+1]}
for(n=0, 25, print1( a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 23 2026
STATUS
approved