cp's OEIS Frontend

A360581 Expansion of A(x) satisfying [x^n] A(x)^n / (1 + x*A(x)^n)^n = 0 for n > 0.

%I A360581 #9 Sep 13 2023 10:49:00
%S A360581 1,1,3,17,131,1204,12587,149131,2036675,32358153,587313706,
%T A360581 11761213199,252859744189,5785648936988,141627609404793,
%U A360581 3737907237793369,106414467836076985,3241492594168333618,104522041356412895455,3541554178675758259947,125782730912626755808358
%N A360581 Expansion of A(x) satisfying [x^n] A(x)^n / (1 + x*A(x)^n)^n = 0 for n > 0.
%H A360581 Paul D. Hanna, <a href="/A360581/b360581.txt">Table of n, a(n) for n = 0..300</a>
%F A360581 From _Vaclav Kotesovec_, Mar 13 2023: (Start)
%F A360581 a(n) ~ c * n! * n^alpha / LambertW(1)^n, where alpha = 0.33953... and c = 0.1881608377753...
%F A360581 Conjecture: alpha = 3*LambertW(1) - 2 + 1/(1 + LambertW(1)) = 0.33953361459446... (End)
%e A360581 G.f. A(x) = 1 + x + 3*x^2 + 17*x^3 + 131*x^4 + 1204*x^5 + 12587*x^6 + 149131*x^7 + 2036675*x^8 + 32358153*x^9 + 587313706*x^10 + ...
%e A360581 The table of coefficients in the successive powers of g.f. A(x) begins:
%e A360581 n = 1: [1, 1,  3,  17,  131,  1204,  12587,  149131, ...];
%e A360581 n = 2: [1, 2,  7,  40,  305,  2772,  28657,  335114, ...];
%e A360581 n = 3: [1, 3, 12,  70,  531,  4782,  48936,  565245, ...];
%e A360581 n = 4: [1, 4, 18, 108,  819,  7324,  74272,  848064, ...];
%e A360581 n = 5: [1, 5, 25, 155, 1180, 10501, 105650, 1193530, ...];
%e A360581 n = 6: [1, 6, 33, 212, 1626, 14430, 144208, 1613214, ...];
%e A360581 n = 7: [1, 7, 42, 280, 2170, 19243, 191254, 2120511, ...];
%e A360581 n = 8: [1, 8, 52, 360, 2826, 25088, 248284, 2730872, ...];
%e A360581 ...
%e A360581 The table of coefficients in A(x)^n/(1 + x*A(x)^n)^n begins:
%e A360581 n = 1: [1, 0,   2,   12,  100,  955, 10258, 124565, ...];
%e A360581 n = 2: [1, 0,   2,   18,  161, 1606, 17757, 220834, ...];
%e A360581 n = 3: [1, 0,   0,   15,  168, 1806, 21000, 272856, ...];
%e A360581 n = 4: [1, 0,  -4,    0,  114, 1504, 19220, 270692, ...];
%e A360581 n = 5: [1, 0, -10,  -30,    0,  800, 12970, 215445, ...];
%e A360581 n = 6: [1, 0, -18,  -78, -165,    0,  4797, 123990, ...];
%e A360581 n = 7: [1, 0, -28, -147, -364, -329,     0,  32767, ...];
%e A360581 n = 8: [1, 0, -40, -240, -572,  696,  7472,      0, ...];
%e A360581 ...
%e A360581 in which the diagonal of all zeros illustrates that
%e A360581 [x^n] A(x)^n / (1 + x*A(x)^n)^n = 0 for n > 0.
%o A360581 (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
%o A360581 A[#A] = -polcoeff( Ser(A)^(#A)/(1 + x*Ser(A)^(#A))^(#A), #A-1)/(#A) );A[n+1]}
%o A360581 for(n=0,30,print1(a(n),", "))
%Y A360581 Cf. A360582, A360583, A360584, A303063.
%K A360581 nonn
%O A360581 0,3
%A A360581 _Paul D. Hanna_, Mar 12 2023