OFFSET
0,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
a(n) ~ c * d^n * n! / n^2, where d = -4 / (LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) = 6.17655460948348035823168... and c = 0.06049920104... - Vaclav Kotesovec, Oct 06 2020
EXAMPLE
G.f.: A(x) = 1 + 2*x + 2*x^2 + 8*x^3 + 94*x^4 + 1754*x^5 + 44638*x^6 + 1430912*x^7 + 55127738*x^8 + 2474943058*x^9 + 126686496842*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients in (1+x)^(n^2) / A(x)^n begins:
n=1: [1, -1, 0, -6, -74, -1500, -39688, -1302742, ...];
n=2: [1, 0, -2, -12, -159, -3136, -82180, -2680752, ...];
n=3: [1, 3, 0, -26, -300, -5454, -137764, -4398210, ...];
n=4: [1, 8, 24, 0, -548, -9576, -223760, -6847536, ...];
n=5: [1, 15, 100, 350, 0, -16022, -376660, -10771830, ...];
n=6: [1, 24, 270, 1844, 7641, 0, -596908, -17643792, ...];
n=7: [1, 35, 588, 6258, 46186, 224196, 0, -26940146, ...];
n=8: [1, 48, 1120, 16864, 182640, 1478160, 8281968, 0, ...]; ...
in which the main diagonal equals all zeros after the initial term, illustrating that [x^(n-1)] (1+x)^(n^2) / A(x)^n = 0 for n>1.
RELATED SERIES.
Define B(x) by [x^(n-1)] A(x)^n / B(x)^(n^2) = 0, then B(x) is an integer series that begins
B(x) = 1 + x + x^2 + x^3 + 33*x^4 + 161*x^5 + 11061*x^6 + 97837*x^7 + 10690245*x^8 + 130880301*x^9 + 20319470589*x^10 + 308187480741*x^11 + 63533886021453*x^12 + 1128112773827877*x^13 + 294884291428281145*x^14 + 5933604178561639785*x^15 + ...
PROG
(PARI) {a(n) = my(A=[1]); for(m=1, n+1, A=concat(A, 0); A[m] = Vec( (1+x +x*O(x^n))^(m^2)/Ser(A)^m )[m]/m ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 17 2018
STATUS
approved