OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..400
FORMULA
a(n) ~ c * 7^(2*n) * n^(n + 17/10) / (exp(n) * 6^n), where c = 0.132594844139368414... - Vaclav Kotesovec, Feb 25 2026
EXAMPLE
O.g.f.: A(x) = 1 + x + 16*x^2 + 480*x^3 + 20094*x^4 + 1051232*x^5 + 65024482*x^6 + 4603101446*x^7 + 365116402526*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n*(n+1)*x*A(x)^7) / A(x)^(n+1) for n >= 0 begins
n = 0: [1, -1, -30, -2694, -454200, -119739480, ...];
n = 1: [1, 0, -34, -3976, -737400, -204152768, ...];
n = 2: [1, 3, 0, -3624, -861480, -260085384, ...];
n = 3: [1, 8, 108, 0, -724248, -279054528, ...];
n = 4: [1, 15, 350, 11030, 0, -225030320, ...];
n = 5: [1, 24, 810, 37896, 2118240, 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)^7)/A(x)^2 = 1 + 0*x - 34*x^2/2! - 3976*x^3/3! - 737400*x^4/4! - 204152768*x^5/5! + ...
n = 2: exp(2*3*x*A(x)^7)/A(x)^3 = 1 + 3*x + 0*x^2/2! - 3624*x^3/3! - 861480*x^4/4! - 260085384*x^5/5! + ...
n = 3: exp(3*4*x*A(x)^7)/A(x)^4 = 1 + 8*x + 108*x^2/2! + 0*x^3/3! - 724248*x^4/4! - 279054528*x^5/5! + ...
n = 4: exp(4*5*x*A(x)^7)/A(x)^5 = 1 + 15*x + 350*x^2/2! + 11030*x^3/3! + 0*x^4/4! - 225030320*x^5/5! + ...
n = 5: exp(5*6*x*A(x)^7)/A(x)^6 = 1 + 24*x + 810*x^2/2! + 37896*x^3/3! + 2118240*x^4/4! + 0*x^5/5! + ...
...
illustrating that [x^n] exp(n*(n+1)*x*A(x)^7) / 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)^7 +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 25 2026
STATUS
approved