%I #35 Oct 20 2020 03:31:38

%S 1,1,1,2,10,78,787,9640,138098,2257718,41409511,841370530,18753127340,

%T 454916534472,11932152340884,336529124983248,10156870523418506,

%U 326668220673870022,11154514687950236767,403044704522955726502,15364691402538733828594,616307960867208404852222,25948893032918425391596907

%N G.f. A(x) satisfies: [x^n] A(x)^n / (x*A(x))' = 0 for n > 1.

%C Odd terms seem to occur only at positions 0, 1, and 2*A118113(k) for k >= 0.

%C Compare to identity: [x^n] (x*F(x))' / F(x)^(n+1) = 0 holds when F(0) = 1.

%C More generally, [x^n] G(x,k)^(k*(n+1)-1) / (x*G(x,k)^k)' = 0 is satisfied by an integer series G(x,k) when k is a fixed positive integer.

%H Paul D. Hanna, <a href="/A300994/b300994.txt">Table of n, a(n) for n = 0..300</a>

%F G.f. A(x) satisfies: [x^n] A(x)^n / (A(x) + x*A'(x)) = 0 for n > 1.

%F a(n) ~ c * 2^n * (n-1)!, where c = 0.1261880758068409567445... - _Vaclav Kotesovec_, Oct 20 2020

%e G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 10*x^4 + 78*x^5 + 787*x^6 + 9640*x^7 + 138098*x^8 + 2257718*x^9 + 41409511*x^10 + ...

%e such that [x^n] A(x)^n / (x*A(x))' = 0 for n > 1.

%e ILLUSTRATION OF DEFINITION.

%e The table of coefficients in A(x)^n / (x*A(x))' begins:

%e n=0: [1, -2, 1, -4, -29, -306, -3892, -57436, -961833, -17990518, ...];

%e n=1: [1, -1, 0, -3, -26, -279, -3594, -53588, -904770, -17034879, ...];

%e n=2: [1, 0, 0, -2, -21, -240, -3196, -48690, -834546, -15887984, ...];

%e n=3: [1, 1, 1, 0, -13, -185, -2674, -42548, -749180, -14525506, ...];

%e n=4: [1, 2, 3, 4, 0, -108, -1997, -34928, -646377, -12919990, ...];

%e n=5: [1, 3, 6, 11, 21, 0, -1124, -25545, -523467, -11040387, ...];

%e n=6: [1, 4, 10, 22, 54, 152, 0, -14048, -377328, -8851500, ...];

%e n=7: [1, 5, 15, 38, 104, 366, 1449, 0, -204288, -6313320, ...];

%e n=8: [1, 6, 21, 60, 177, 666, 3322, 17148, 0, -3380224, ...];

%e n=9: [1, 7, 28, 89, 280, 1083, 5750, 38090, 240717, 0, ...]; ...

%e in which the main diagonal consists of all zeros after the initial terms, illustrating that [x^n] A(x)^n / (x*A(x))' = 0 for n > 1.

%e RELATED SERIES.

%e (x*A(x))' = 1 + 2*x + 3*x^2 + 8*x^3 + 50*x^4 + 468*x^5 + 5509*x^6 + 77120*x^7 + 1242882*x^8 + 22577180*x^9 + 455504621*x^10 + ...

%e log(A(x)) = x + x^2/2 + 4*x^3/3 + 33*x^4/4 + 341*x^5/5 + 4252*x^6/6 + 61916*x^7/7 + 1026865*x^8/8 + 19065307*x^9/9 + 391349641*x^10/10 + ...

%o (PARI) {a(n) = my(A=[1,1]); for(i=1,n, A = concat(A,0); A[#A] = Vec( Ser(A)^(#A-1)/(x*Ser(A))' )[#A]); A[n+1]}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A300995, A300627, A302059, A302060.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Mar 29 2018