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A302703 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n+1))^(n+1) for n>=0.

%I A302703 #24 Feb 06 2023 11:27:16
%S A302703 1,1,3,21,235,3470,61933,1274893,29423331,747440115,20636072811,
%T A302703 613611700946,19517927805840,660667692682175,23699856058131981,
%U A302703 897955765812058192,35832679277251514074,1502303284645831488072,66031982339561373164915,3036884343153028302140119,145885192794643951791449387
%N A302703 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 A302703 Paul D. Hanna, <a href="/A302703/b302703.txt">Table of n, a(n) for n = 0..300</a>
%F A302703 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
%F A302703 (1) [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n+1))^(n+1) for n>=0.
%F A302703 (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 A302703 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 A302703 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 A302703 RELATED SERIES.
%e A302703 G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
%e A302703 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 A302703 such that b(n) = [x^n] (1 + x*A(x)^(n+1))^(n+1) / (n+1),
%e A302703 as well as b(n) = [x^n] A(x)^(n+1) / (n+1),
%e A302703 so that b(n) begin:
%e A302703 [1, 2/2, 12/3, 124/4, 1780/5, 31746/6, 662144/7, 15590104/8, ...]
%e A302703 ILLUSTRATION OF DEFINITION.
%e A302703 The table of coefficients of x^k in A(x)^(n+1) begins:
%e A302703 n=0: [1, 1,  3,  21,  235,  3470,  61933,  1274893, ...];
%e A302703 n=1: [1, 2,  7,  48,  521,  7536, 132657,  2704342, ...];
%e A302703 n=2: [1, 3, 12,  82,  867, 12288, 213282,  4304877, ...];
%e A302703 n=3: [1, 4, 18, 124, 1283, 17828, 305056,  6094832, ...];
%e A302703 n=4: [1, 5, 25, 175, 1780, 24271, 409380,  8094540, ...];
%e A302703 n=5: [1, 6, 33, 236, 2370, 31746, 527824, 10326546, ...];
%e A302703 n=6: [1, 7, 42, 308, 3066, 40397, 662144, 12815839, ...];
%e A302703 n=7: [1, 8, 52, 392, 3882, 50384, 814300, 15590104, ...]; ...
%e A302703 Compare to the table of coefficients in (1 + x*A(x)^(n+1))^(n+1):
%e A302703 n=0: [1, 1,  1,   3,   21,   235,    3470,    61933, ...];
%e A302703 n=1: [1, 2,  5,  18,  114,  1166,   16355,   283142, ...];
%e A302703 n=2: [1, 3, 12,  55,  354,  3372,   44463,   739917, ...];
%e A302703 n=3: [1, 4, 22, 124,  857,  7908,   98244,  1558788, ...];
%e A302703 n=4: [1, 5, 35, 235, 1780, 16501,  195980,  2955095, ...];
%e A302703 n=5: [1, 6, 51, 398, 3321, 31746,  368032,  5294250, ...];
%e A302703 n=6: [1, 7, 70, 623, 5719, 57302,  662144,  9182013, ...];
%e A302703 n=7: [1, 8, 92, 920, 9254, 98088, 1149804, 15590104, ...]; ...
%e A302703 to see that the main diagonals of the tables are the same.
%o A302703 (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 A302703 for(n=0, 30, print1(a(n), ", "))
%Y A302703 Cf. A360231, A302702, A360234, A360235, A360236, A360237.
%Y A302703 Cf. A360345, A360337.
%K A302703 nonn
%O A302703 0,3
%A A302703 _Paul D. Hanna_, Apr 16 2018