|
%I #13 Dec 03 2022 12:05:27
%S 1,1,3,37,1083,53701,3934443,395502997,51998075643,8643190760101,
%T 1770707733052683,438247927304923957,128926370847248904603,
%U 44477192002157773868101,17787359321176954021105323,8164879801560415752441320917,4264616618784184682736855291963
%N a(n) = coefficient of x^n/n!, n >= 0, in A(x) such that: 0 = Sum_{n>=1} exp(-n^2*x) * (exp(n*x) - A(x))^n.
%C All terms appear to be odd.
%C Conjecture: after initial term, sequence modulo 10 has period 4: [1,3,7,3] repeating.
%H Paul D. Hanna, <a href="/A357398/b357398.txt">Table of n, a(n) for n = 0..200</a>
%e G.f.: A(x) = 1 + x + 3*x^2/2! + 37*x^3/3! + 1083*x^4/4! + 53701*x^5/5! + 3934443*x^6/6! + 395502997*x^7/7! + 51998075643*x^8/8! + 8643190760101*x^9/9! + 1770707733052683*x^10/10! + ...
%e where
%e 0 = exp(-x)*(exp(x) - A(x)) + exp(-4*x)*(exp(2*x) - A(x))^2 + exp(-9*x)*(exp(3*x) - A(x))^3 + exp(-16*x)*(exp(4*x) - A(x))^4 + exp(-25*x)*(exp(5*x) - A(x))^5 + ... + exp(-n^2*x) * (exp(x)^n - A(x))^n + ...
%e equivalently,
%e 0 = (1 - exp(-x)*A(x)) + (1 - exp(-2*x)*A(x))^2 + (1 - exp(-3*x)*A(x))^3 + (1 - exp(-4*x)*A(x))^4 + (1 - exp(-5*x)*A(x))^5 + ... + (1 - exp(-n*x)*A(x))^n + ...
%e RELATED SERIES.
%e log(A(x)) = x + 2*x^2/2! + 30*x^3/3! + 938*x^4/4! + 48030*x^5/5! + 3590522*x^6/6! + 366038190*x^7/7! + 48627434858*x^8/8! + 8147891495550*x^9/9! + 1679856055853402*x^10/10! + ...
%e Conjecture: after initial term, the coefficients of log(A(x)) modulo 10 has period 4: [2,0,8,0] repeating.
%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
%o A[#A] = polcoeff( sum(m=1,#A-1, (1 - Ser(A)*exp(-m*x +x*O(x^#A)))^m ),#A-1) ); n!*A[n+1]}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A357397.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Oct 20 2022
| 