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A360584 Expansion of A(x) satisfying [x^n] A(x) / (1 + x*A(x)^(n+2)) = 0 for n > 0.

%I A360584 #9 Mar 13 2023 04:26:27
%S A360584 1,1,4,29,294,3727,55748,950898,18094313,378363501,8600306451,
%T A360584 210773059751,5534376088000,154911828439188,4603267204022882,
%U A360584 144710918709587399,4798300212740184379,167370947204751098624,6127130537038980726113,234905895680130694945861,9413383171884998924237972
%N A360584 Expansion of A(x) satisfying [x^n] A(x) / (1 + x*A(x)^(n+2)) = 0 for n > 0.
%H A360584 Paul D. Hanna, <a href="/A360584/b360584.txt">Table of n, a(n) for n = 0..300</a>
%F A360584 a(n) ~ c * n! * n^(4*LambertW(1) - 1 + 2/(1 + LambertW(1))) / LambertW(1)^n, where c = 0.02048373460253911846... - _Vaclav Kotesovec_, Mar 13 2023
%e A360584 G.f.: A(x) = 1 + x + 4*x^2 + 29*x^3 + 294*x^4 + 3727*x^5 + 55748*x^6 + 950898*x^7 + 18094313*x^8 + 378363501*x^9 + 8600306451*x^10 + ...
%e A360584 The table of coefficients in the successive powers of g.f. A(x) begins:
%e A360584 n = 1: [1, 1,  4,  29,  294,  3727,  55748, ...];
%e A360584 n = 2: [1, 2,  9,  66,  662,  8274, 122143, ...];
%e A360584 n = 3: [1, 3, 15, 112, 1116, 13776, 200827, ...];
%e A360584 n = 4: [1, 4, 22, 168, 1669, 20384, 293654, ...];
%e A360584 n = 5: [1, 5, 30, 235, 2335, 28266, 402710, ...];
%e A360584 n = 6: [1, 6, 39, 314, 3129, 37608, 530334, ...];
%e A360584 n = 7: [1, 7, 49, 406, 4067, 48615, 679140, ...];
%e A360584 ...
%e A360584 The table of coefficients in A(x)/(1 + x*A(x)^(n+2)) begins:
%e A360584 n = 1: [1, 0,  1,  13,  166,  2391,  38776, 699060, ...];
%e A360584 n = 2: [1, 0,  0,   7,  119,  1911,  32823, 612983, ...];
%e A360584 n = 3: [1, 0, -1,   0,   64,  1358,  26039, 515774, ...];
%e A360584 n = 4: [1, 0, -2,  -8,    0,   724,  18356, 406634, ...];
%e A360584 n = 5: [1, 0, -3, -17,  -74,     0,   9702, 284785, ...];
%e A360584 n = 6: [1, 0, -4, -27, -159,  -824,      0, 149478, ...];
%e A360584 n = 7: [1, 0, -5, -38, -256, -1759, -10833,      0, ...];
%e A360584 ...
%e A360584 in which the diagonal of all zeros illustrates that
%e A360584 [x^n] A(x) / (1 + x*A(x)^(n+2)) = 0 for n > 0.
%o A360584 (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
%o A360584 A[#A] = -polcoeff( Ser(A)/(1 + x*Ser(A)^(#A+1)), #A-1) );A[n+1]}
%o A360584 for(n=0,30,print1(a(n),", "))
%Y A360584 Cf. A360582, A360583.
%K A360584 nonn
%O A360584 0,3
%A A360584 _Paul D. Hanna_, Mar 12 2023