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A360582 Expansion of A(x) satisfying [x^n] A(x) / (1 + x*A(x)^n) = 0 for n > 0.

%I A360582 #11 Mar 13 2023 04:05:19
%S A360582 1,1,2,8,48,382,3793,45208,627957,9928646,175476102,3420270423,
%T A360582 72789704826,1678446235555,41675807453127,1108522434288617,
%U A360582 31444611938560078,947522959703143140,30225484159719768548,1017558928058932606182,36053690169955373601165,1341103168079733579768368
%N A360582 Expansion of A(x) satisfying [x^n] A(x) / (1 + x*A(x)^n) = 0 for n > 0.
%H A360582 Paul D. Hanna, <a href="/A360582/b360582.txt">Table of n, a(n) for n = 0..300</a>
%F A360582 a(n) ~ c * n! * n^(2*LambertW(1) - 1) / LambertW(1)^n, where c = 0.11249164340900724981958... - _Vaclav Kotesovec_, Mar 13 2023
%e A360582 G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 48*x^4 + 382*x^5 + 3793*x^6 + 45208*x^7 + 627957*x^8 + 9928646*x^9 + 175476102*x^10 + ...
%e A360582 The table of coefficients in the successive powers of g.f. A(x) begins:
%e A360582 n = 1: [1, 1,  2,   8,   48,  382,  3793,  45208, ...];
%e A360582 n = 2: [1, 2,  5,  20,  116,  892,  8606, 100298, ...];
%e A360582 n = 3: [1, 3,  9,  37,  210, 1566, 14687, 167280, ...];
%e A360582 n = 4: [1, 4, 14,  60,  337, 2448, 22340, 248580, ...];
%e A360582 n = 5: [1, 5, 20,  90,  505, 3591, 31935, 347120, ...];
%e A360582 n = 6: [1, 6, 27, 128,  723, 5058, 43919, 466410, ...];
%e A360582 n = 7: [1, 7, 35, 175, 1001, 6923, 58828, 610653, ...];
%e A360582 ...
%e A360582 The table of coefficients in A(x)/(1 + x*A(x)^n) begins:
%e A360582 n = 1: [1, 0,  1,   5,  34,  293, 3066, 37900, ...];
%e A360582 n = 2: [1, 0,  0,   3,  25,  235, 2601, 33346, ...];
%e A360582 n = 3: [1, 0, -1,   0,  14,  167, 2055, 28049, ...];
%e A360582 n = 4: [1, 0, -2,  -4,   0,   89, 1432, 21994, ...];
%e A360582 n = 5: [1, 0, -3,  -9, -18,    0,  742, 15216, ...];
%e A360582 n = 6: [1, 0, -4, -15, -41, -102,    0,  7820, ...];
%e A360582 n = 7: [1, 0, -5, -22, -70, -220, -775,     0, ...];
%e A360582 ...
%e A360582 in which the diagonal of all zeros illustrates that
%e A360582 [x^n] A(x) / (1 + x*A(x)^n) = 0 for n > 0.
%o A360582 (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
%o A360582 A[#A] = -polcoeff( Ser(A)/(1 + x*Ser(A)^(#A-1)), #A-1) );A[n+1]}
%o A360582 for(n=0,30,print1(a(n),", "))
%Y A360582 Cf. A360581, A360583, A360584.
%K A360582 nonn
%O A360582 0,3
%A A360582 _Paul D. Hanna_, Mar 12 2023