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A360234 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n+2))^(n+1) for n >= 0.

%I A360234 #19 Feb 06 2023 11:28:07
%S A360234 1,1,4,33,414,6750,131963,2957899,73968136,2027178710,60143834893,
%T A360234 1914750144642,64984397381766,2339387034919340,88976089246855623,
%U A360234 3563952072597604091,149941204887915187568,6610797722288579969347,304837386103152855175255,14675559490665539299350303
%N A360234 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n+2))^(n+1) for n >= 0.
%H A360234 Paul D. Hanna, <a href="/A360234/b360234.txt">Table of n, a(n) for n = 0..300</a>
%F A360234 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
%F A360234 (1) [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n+2))^(n+1) for n>=0.
%F A360234 (2) A(x) = Sum_{n>=0} b(n) * x^n/A(x)^n, where b(n) = [x^n] (1 + x*A(x)^(n+2))^(n+1) / (n+1).
%F A360234 a(n) ~ c * d^n * n! * n^alfa, where d = A360279 = 2.12460658362428979..., alfa = 3.146325060582260657459991059461810..., c = 0.007037477865521004701131626931596125... - _Vaclav Kotesovec_, Jan 31 2023
%e A360234 G.f.: A(x) = 1 + x + 4*x^2 + 33*x^3 + 414*x^4 + 6750*x^5 + 131963*x^6 + 2957899*x^7 + 73968136*x^8 + 2027178710*x^9 + 60143834893*x^10 + ...
%e A360234 RELATED SERIES.
%e A360234 G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
%e A360234 B(x) = 1 + x + 5*x^2 + 46*x^3 + 603*x^4 + 10011*x^5 + 197357*x^6 + 4444483*x^7 + 111520277*x^8 + ... + b(n)*x^n + ...
%e A360234 such that b(n) = [x^n] (1 + x*A(x)^(n+2))^(n+1) / (n+1),
%e A360234 as well as b(n) = [x^n] A(x)^(n+1) / (n+1),
%e A360234 so that b(n) begin:
%e A360234 [1/1, 2/2, 15/3, 184/4, 3015/5, 60066/6, 1381499/7, 35555864/8, ...].
%e A360234 ILLUSTRATION OF DEFINITION.
%e A360234 The table of coefficients of x^k in A(x)^(n+1) begins:
%e A360234 n=0: [1, 1,  4,  33,  414,  6750,  131963,  2957899, ...];
%e A360234 n=1: [1, 2,  9,  74,  910, 14592,  281827,  6261048, ...];
%e A360234 n=2: [1, 3, 15, 124, 1500, 23673,  451690,  9944484, ...];
%e A360234 n=3: [1, 4, 22, 184, 2197, 34156,  643878, 14046740, ...];
%e A360234 n=4: [1, 5, 30, 255, 3015, 46221,  860965, 18610170, ...];
%e A360234 n=5: [1, 6, 39, 338, 3969, 60066, 1105794, 23681298, ...];
%e A360234 n=6: [1, 7, 49, 434, 5075, 75908, 1381499, 29311192, ...];
%e A360234 n=7: [1, 8, 60, 544, 6350, 93984, 1691528, 35555864, ...]; ...
%e A360234 Compare to the table of coefficients in (1 + x*A(x)^(n+2))^(n+1):
%e A360234 n=0: [1, 1,   2,    9,    74,    910,   14592,   281827, ...];
%e A360234 n=1: [1, 2,   7,   36,   287,   3338,   51315,   963446, ...];
%e A360234 n=2: [1, 3,  15,   91,   744,   8337,  122662,  2227101, ...];
%e A360234 n=3: [1, 4,  26,  184,  1591,  17600,  249194,  4361112, ...];
%e A360234 n=4: [1, 5,  40,  325,  3015,  33656,  463710,  7824385, ...];
%e A360234 n=5: [1, 6,  57,  524,  5244,  60066,  816474, 13339956, ...];
%e A360234 n=6: [1, 7,  77,  791,  8547, 101619, 1381499, 22023891, ...];
%e A360234 n=7: [1, 8, 100, 1136, 13234, 164528, 2263888, 35555864, ...]; ...
%e A360234 to see that the main diagonals of the tables are the same.
%o A360234 (PARI) {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(m+2))^(m+1))[m+1] - Vec(Ser(A)^(m+1))[m+1])/(m+1) ); A[n+1]}
%o A360234 for(n=0, 20, print1(a(n), ", "))
%Y A360234 Cf. A360231, A302702, A302703, A360235, A360236, A360237.
%Y A360234 Cf. A360346, A360338.
%K A360234 nonn
%O A360234 0,3
%A A360234 _Paul D. Hanna_, Jan 30 2023