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Expansion of A(x) satisfying [x^n] A(x) / (1 + x*A(x)^n) = 0 for n > 0.
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%I #11 Mar 13 2023 04:05:19

%S 1,1,2,8,48,382,3793,45208,627957,9928646,175476102,3420270423,

%T 72789704826,1678446235555,41675807453127,1108522434288617,

%U 31444611938560078,947522959703143140,30225484159719768548,1017558928058932606182,36053690169955373601165,1341103168079733579768368

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

%H Paul D. Hanna, <a href="/A360582/b360582.txt">Table of n, a(n) for n = 0..300</a>

%F a(n) ~ c * n! * n^(2*LambertW(1) - 1) / LambertW(1)^n, where c = 0.11249164340900724981958... - _Vaclav Kotesovec_, Mar 13 2023

%e 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 The table of coefficients in the successive powers of g.f. A(x) begins:

%e n = 1: [1, 1, 2, 8, 48, 382, 3793, 45208, ...];

%e n = 2: [1, 2, 5, 20, 116, 892, 8606, 100298, ...];

%e n = 3: [1, 3, 9, 37, 210, 1566, 14687, 167280, ...];

%e n = 4: [1, 4, 14, 60, 337, 2448, 22340, 248580, ...];

%e n = 5: [1, 5, 20, 90, 505, 3591, 31935, 347120, ...];

%e n = 6: [1, 6, 27, 128, 723, 5058, 43919, 466410, ...];

%e n = 7: [1, 7, 35, 175, 1001, 6923, 58828, 610653, ...];

%e ...

%e The table of coefficients in A(x)/(1 + x*A(x)^n) begins:

%e n = 1: [1, 0, 1, 5, 34, 293, 3066, 37900, ...];

%e n = 2: [1, 0, 0, 3, 25, 235, 2601, 33346, ...];

%e n = 3: [1, 0, -1, 0, 14, 167, 2055, 28049, ...];

%e n = 4: [1, 0, -2, -4, 0, 89, 1432, 21994, ...];

%e n = 5: [1, 0, -3, -9, -18, 0, 742, 15216, ...];

%e n = 6: [1, 0, -4, -15, -41, -102, 0, 7820, ...];

%e n = 7: [1, 0, -5, -22, -70, -220, -775, 0, ...];

%e ...

%e in which the diagonal of all zeros illustrates that

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

%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);

%o A[#A] = -polcoeff( Ser(A)/(1 + x*Ser(A)^(#A-1)), #A-1) );A[n+1]}

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

%Y Cf. A360581, A360583, A360584.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Mar 12 2023