%I #26 Feb 05 2023 03:14:38
%S 1,1,2,12,120,1595,25823,485254,10278756,240814116,6159248281,
%T 170371486813,5060981349876,160573684489465,5417789356278015,
%U 193693975380448414,7315287863625954712,291082028021247460862,12174286414586563087259,534059044249856004891501,24524697505864740171996008
%N G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^n)^(n+1) for n>=0.
%H Paul D. Hanna, <a href="/A302702/b302702.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)^(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)^(n+1) / (n+1).
%F a(n) ~ c * d^n * n! * n^alfa, where d = A360279 = 2.1246065836242897918278825..., alfa = 1.256334309718765863868089027485828533429844901971596190707510781..., c = 0.080161548550419985236395573058502044572123359124998971614... - _Vaclav Kotesovec_, Oct 06 2020, updated Feb 05 2023
%e G.f.: A(x) = 1 + x + 2*x^2 + 12*x^3 + 120*x^4 + 1595*x^5 + 25823*x^6 + 485254*x^7 + 10278756*x^8 + 240814116*x^9 + 6159248281*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 + 3*x^2 + 19*x^3 + 189*x^4 + 2496*x^5 + 40216*x^6 + 753775*x^7 + 15956057*x^8 + 374080591*x^9 + 6159248281*x^10 + ... + b(n)*x^n + ...
%e such that b(n) = [x^n] (1 + x*A(x)^n)^(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, 9/3, 76/4, 945/5, 14976/6, 281512/7, 6030200/8, ...]
%e ILLUSTRATION OF DEFINITION.
%e The table of coefficients of x^k in A(x)^(n+1) begins:
%e n=0: [1, 1, 2, 12, 120, 1595, 25823, 485254, ...];
%e n=1: [1, 2, 5, 28, 268, 3478, 55460, 1031414, ...];
%e n=2: [1, 3, 9, 49, 450, 5697, 89423, 1645281, ...];
%e n=3: [1, 4, 14, 76, 673, 8308, 128296, 2334456, ...];
%e n=4: [1, 5, 20, 110, 945, 11376, 172745, 3107440, ...];
%e n=5: [1, 6, 27, 152, 1275, 14976, 223529, 3973746, ...];
%e n=6: [1, 7, 35, 203, 1673, 19194, 281512, 4944024, ...];
%e n=7: [1, 8, 44, 264, 2150, 24128, 347676, 6030200, ...]; ...
%e Compare to the table of coefficients in (1 + x*A(x)^n)^(n+1):
%e n=0: [1, 1, 0, 0, 0, 0, 0, 0, ...];
%e n=1: [1, 2, 3, 6, 29, 268, 3458, 55124, ...];
%e n=2: [1, 3, 9, 28, 132, 1059, 12605, 192579, ...];
%e n=3: [1, 4, 18, 76, 395, 2940, 31872, 459048, ...];
%e n=4: [1, 5, 30, 160, 945, 6986, 70100, 940180, ...];
%e n=5: [1, 6, 45, 290, 1950, 14976, 143807, 1796430, ...];
%e n=6: [1, 7, 63, 476, 3619, 29589, 281512, 3321571, ...];
%e n=7: [1, 8, 84, 728, 6202, 54600, 529116, 6030200, ...]; ...
%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)^(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, A303062, A302703, A360234, A360235, A360236, A360237.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Apr 16 2018