OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..400
FORMULA
a(n) ~ c * 5^(2*n) * n^(n + 11/6) / (2^(2*n) * exp(n)), where c = 0.15773058... - Vaclav Kotesovec, Feb 25 2026
EXAMPLE
O.g.f.: A(x) = 1 + x + 12*x^2 + 271*x^3 + 8620*x^4 + 345505*x^5 + 16482108*x^6 + 904341918*x^7 + 55798640932*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n*(n+1)*x*A(x)^5) / A(x)^(n+1) for n >= 0 begins
n = 0: [1, -1, -22, -1488, -191256, -38755080, ...];
n = 1: [1, 0, -26, -2272, -318792, -67499968, ...];
n = 2: [1, 3, 0, -2202, -391608, -89502624, ...];
n = 3: [1, 8, 92, 0, -354168, -102027648, ...];
n = 4: [1, 15, 310, 7820, 0, -89284720, ...];
n = 5: [1, 24, 738, 28752, 1245744, 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)^5)/A(x)^2 = 1 + 0*x - 26*x^2/2! - 2272*x^3/3! - 318792*x^4/4! - 67499968*x^5/5! + ...
n = 2: exp(2*3*x*A(x)^5)/A(x)^3 = 1 + 3*x + 0*x^2/2! - 2202*x^3/3! - 391608*x^4/4! - 89502624*x^5/5! + ...
n = 3: exp(3*4*x*A(x)^5)/A(x)^4 = 1 + 8*x + 92*x^2/2! + 0*x^3/3! - 354168*x^4/4! - 102027648*x^5/5! + ...
n = 4: exp(4*5*x*A(x)^5)/A(x)^5 = 1 + 15*x + 310*x^2/2! + 7820*x^3/3! + 0*x^4/4! - 89284720*x^5/5! + ...
n = 5: exp(5*6*x*A(x)^5)/A(x)^6 = 1 + 24*x + 738*x^2/2! + 28752*x^3/3! + 1245744*x^4/4! + 0*x^5/5! + ...
...
illustrating that [x^n] exp(n*(n+1)*x*A(x)^5) / 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)^5 +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