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 = 0.8017341554... - Vaclav Kotesovec, Feb 24 2026
EXAMPLE
O.g.f.: A(x) = 1 + x + 4*x^2 + 33*x^3 + 448*x^4 + 8782*x^5 + 227824*x^6 + 7370601*x^7 + 285458448*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n*(n+1)*x*A(x)) / A(x)^(n+1) for n >= 0 begins
n = 0: [1, -1, -6, -156, -9048, -930480, ...];
n = 1: [1, 0, -10, -304, -17832, -1841408, ...];
n = 2: [1, 3, 0, -438, -29592, -3017304, ...];
n = 3: [1, 8, 60, 0, -40632, -4746048, ...];
n = 4: [1, 15, 230, 3200, 0, -6368120, ...];
n = 5: [1, 24, 594, 14784, 327312, 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))/A(x)^2 = 1 + 0*x - 10*x^2/2! - 304*x^3/3! - 17832*x^4/4! - 1841408*x^5/5! + ...
n = 2: exp(2*3*x*A(x))/A(x)^3 = 1 + 3*x + 0*x^2/2! - 438*x^3/3! - 29592*x^4/4! - 3017304*x^5/5! + ...
n = 3: exp(3*4*x*A(x))/A(x)^4 = 1 + 8*x + 60*x^2/2! + 0*x^3/3! - 40632*x^4/4! - 4746048*x^5/5! + ...
n = 4: exp(4*5*x*A(x))/A(x)^5 = 1 + 15*x + 230*x^2/2! + 3200*x^3/3! + 0*x^4/4! - 6368120*x^5/5! + ...
n = 5: exp(5*6*x*A(x))/A(x)^6 = 1 + 24*x + 594*x^2/2! + 14784*x^3/3! + 327312*x^4/4! + 0*x^5/5! + ...
...
illustrating that [x^n] exp(n*(n+1)*x*A(x)) / 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) +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