%I #24 Feb 06 2023 11:27:16

%S 1,1,3,21,235,3470,61933,1274893,29423331,747440115,20636072811,

%T 613611700946,19517927805840,660667692682175,23699856058131981,

%U 897955765812058192,35832679277251514074,1502303284645831488072,66031982339561373164915,3036884343153028302140119,145885192794643951791449387

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

%H Paul D. Hanna, <a href="/A302703/b302703.txt">Table of n, a(n) for n = 0..300</a>

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:

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

%F (2) A(x) = Sum_{n>=0} b(n) * x^n/A(x)^n, where b(n) = [x^n] (1 + x*A(x)^(n+1))^(n+1) / (n+1).

%F a(n) ~ c * d^n * n! * n^alfa, where d = A360279 = 2.1246065836242897918278825..., alfa = 2.2013296851505132606640400434738193121994558898350865326..., c = 0.026186121837027622395555466054900245177877028741031867... - _Vaclav Kotesovec_, Oct 06 2020, updated Feb 05 2023

%e G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 235*x^4 + 3470*x^5 + 61933*x^6 + 1274893*x^7 + 29423331*x^8 + 747440115*x^9 + 20636072811*x^10 + ...

%e RELATED SERIES.

%e G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:

%e B(x) = 1 + x + 4*x^2 + 31*x^3 + 356*x^4 + 5291*x^5 + 94592*x^6 + 1948763*x^7 + 45025516*x^8 + 1145651239*x^9 + 31696223593*x^10 + ... + b(n)*x^n + ...

%e such that b(n) = [x^n] (1 + x*A(x)^(n+1))^(n+1) / (n+1),

%e as well as b(n) = [x^n] A(x)^(n+1) / (n+1),

%e so that b(n) begin:

%e [1, 2/2, 12/3, 124/4, 1780/5, 31746/6, 662144/7, 15590104/8, ...]

%e ILLUSTRATION OF DEFINITION.

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

%e n=0: [1, 1, 3, 21, 235, 3470, 61933, 1274893, ...];

%e n=1: [1, 2, 7, 48, 521, 7536, 132657, 2704342, ...];

%e n=2: [1, 3, 12, 82, 867, 12288, 213282, 4304877, ...];

%e n=3: [1, 4, 18, 124, 1283, 17828, 305056, 6094832, ...];

%e n=4: [1, 5, 25, 175, 1780, 24271, 409380, 8094540, ...];

%e n=5: [1, 6, 33, 236, 2370, 31746, 527824, 10326546, ...];

%e n=6: [1, 7, 42, 308, 3066, 40397, 662144, 12815839, ...];

%e n=7: [1, 8, 52, 392, 3882, 50384, 814300, 15590104, ...]; ...

%e Compare to the table of coefficients in (1 + x*A(x)^(n+1))^(n+1):

%e n=0: [1, 1, 1, 3, 21, 235, 3470, 61933, ...];

%e n=1: [1, 2, 5, 18, 114, 1166, 16355, 283142, ...];

%e n=2: [1, 3, 12, 55, 354, 3372, 44463, 739917, ...];

%e n=3: [1, 4, 22, 124, 857, 7908, 98244, 1558788, ...];

%e n=4: [1, 5, 35, 235, 1780, 16501, 195980, 2955095, ...];

%e n=5: [1, 6, 51, 398, 3321, 31746, 368032, 5294250, ...];

%e n=6: [1, 7, 70, 623, 5719, 57302, 662144, 9182013, ...];

%e n=7: [1, 8, 92, 920, 9254, 98088, 1149804, 15590104, ...]; ...

%e to see that the main diagonals of the tables are the same.

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

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

%Y Cf. A360231, A302702, A360234, A360235, A360236, A360237.

%Y Cf. A360345, A360337.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Apr 16 2018