OFFSET
1,2
COMMENTS
It is remarkable that this sequence should consist entirely of integers.
EXAMPLE
O.g.f.: A(x) = x + 2*x^2 + 27*x^3 + 2176*x^4 + 316125*x^5 + 92433420*x^6 + 38689900249*x^7 + 24036220587520*x^8 + 19705732103751309*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(-n^3*A(x)) / (1 - n^2*x)^n begins:
n=1: [1, 0, -3, -160, -52191, -37930176, -66549456875, ...];
n=2: [1, 0, 0, -1040, -414720, -303430848, -532404700160, ...];
n=3: [1, 0, 135, 0, -1237275, -1019993472, -1799293659165, ...];
n=4: [1, 0, 768, 22400, 0, -2155144704, -4259850874880, ...];
n=5: [1, 0, 2625, 136000, 25862625, 0, -7511859284375, ...];
n=6: [1, 0, 6912, 524880, 192513024, 36792874944, 0, ...];
n=7: [1, 0, 15435, 1591520, 938926485, 280095248832, 121196964253015, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(-m^3*x*Ser(A))/(1-m^2*x +x^2*O(x^m))^m)[m+1]/m^3 ); A[n]}
for(n=1, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 19 2018
STATUS
approved