A230218 G.f. A(x) satisfies: [x^n] A(x)^(n^2+n+1) = 0 for n>1.
1, 1, -3, 14, -85, 504, -4424, 6796, -878157, -25703710, -1270518018, -65772588300, -3848787714746, -248212765567326, -17520121174143210, -1343050785659060872, -111112550557260635229, -9867409274482580015370, -936234289413196544207234, -94522404087905722536648780
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x - 3*x^2 + 14*x^3 - 85*x^4 + 504*x^5 - 4424*x^6 +... Coefficients of x^k in the powers A(x)^(n^2+n+1) of g.f. A(x) begin: n=0: [1, 1, -3, 14, -85, 504, -4424, 6796, ...]; n=1: [1, 3, -6, 25, -153, 819, -8664, -18360, ...]; n=2: [1, 7, 0, 7, -98, 210, -10122, -141525, ...]; n=3: [1,13, 39, 0, -78, -819, -15483, -380952, ...]; n=4: [1,21, 147, 364, 0, -2457, -35805, -821916, ...]; n=5: [1,31, 372, 2139, 5580, 0, -91698, -1792947, ...]; n=6: [1,43, 774, 7525, 42097, 125517, 0, -4097298, ...]; n=7: [1,57, 1425, 20482, 185877, 1089270, 3791298, 0, ...]; n=8: [1,73, 2409, 47450, 619697, 5619978, 35621518, 144591976, 0, ...]; ... where the coefficients of x^n in A(x)^(n^2+n+1) all equal zero for n>1. ODD TERMS: For n>0, a(n) appears to be odd only when n is a power of 2: a(1) = 1; a(2) = -3; a(4) = -85; a(8) = -878157; a(16) = -111112550557260635229; a(32) = -886203693344229341179357569730608605545213045330679133; ...
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..520
Programs
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PARI
{a(n)=local(A=[1,1]); for(m=2,n, A=concat(A, 0); A[#A]=-Vec(Ser(A)^(m^2+m+1))[m+1]/(m^2+m+1)); A[n+1]} for(n=0,20,print1(a(n),", "))
Formula
for n>0, a(n) is odd iff n is a power of 2 (conjecture).
G.f. A(x) satisfies:
(1) A(x) = F(x/A(x)) where F(x) = A(x*F(x)) is the g.f. of A185072.
(2) A(x) = G(x/A(x)^2) where G(x) = A(x*G(x)^2) is the g.f. of A229041.
a(n) ~ -c * 2^(2*n) *n^(n-5/2) / (exp(n) * d^n * (2-d)^n), where d = -LambertW(-2*exp(-2)) = -A226775 = 0.40637573995995990767695812412483975821... and c = 0.015106126717978... - Vaclav Kotesovec, Sep 27 2017