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A360337 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(3*n+1))^(n+1) for n >= 0.

%I A360337 #8 Feb 06 2023 12:09:03
%S A360337 1,1,7,124,3446,125706,5540958,282129207,16148101259,1020687876920,
%T A360337 70377734170699,5246775452965364,420104327765022458,
%U A360337 35937961751407922101,3270668852260460283730,315546031669853942486219,32173855061751806476275665,3457696770952845858846954590
%N A360337 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(3*n+1))^(n+1) for n >= 0.
%H A360337 Paul D. Hanna, <a href="/A360337/b360337.txt">Table of n, a(n) for n = 0..300</a>
%F A360337 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
%F A360337 (1) [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(3*n+1))^(n+1) for n>=0.
%F A360337 (2) A(x) = Sum_{n>=0} b(n) * x^n/A(x)^n, where b(n) = [x^n] (1 + x*A(x)^(3*n+1))^(n+1) / (n+1).
%F A360337 a(n) ~ c * d^n * n! * n^alpha, where d = 5.7189630165873859806..., alpha = 1.4141427006501..., c = 0.027880568114272... - _Vaclav Kotesovec_, Feb 06 2023
%e A360337 G.f.: A(x) = 1 + x + 7*x^2 + 124*x^3 + 3446*x^4 + 125706*x^5 + 5540958*x^6 + 282129207*x^7 + 16148101259*x^8 + 1020687876920*x^9 + ...
%e A360337 RELATED SERIES.
%e A360337 G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
%e A360337 B(x) = 1 + x + 8*x^2 + 146*x^3 + 4083*x^4 + 149077*x^5 + 6569555*x^6 + 334401750*x^7 + 19137707066*x^8 + ... + b(n)*x^n + ...
%e A360337 such that b(n) = [x^n] (1 + x*A(x)^(3*n+1))^(n+1) / (n+1),
%e A360337 as well as b(n) = [x^n] A(x)^(n+1) / (n+1),
%e A360337 so that b(n) begin:
%e A360337 [1/1, 2/2, 24/3, 584/4, 20415/5, 894462/6, 45986885/7, 2675214000/8, ...].
%e A360337 ILLUSTRATION OF DEFINITION.
%e A360337 The table of coefficients of x^k in A(x)^(n+1) begins:
%e A360337 n=0: [1, 1,  7,  124,  3446,  125706,  5540958,  282129207, ...];
%e A360337 n=1: [1, 2, 15,  262,  7189,  260040, 11396948,  577954822, ...];
%e A360337 n=2: [1, 3, 24,  415, 11250,  403521, 17583859,  888063051, ...];
%e A360337 n=3: [1, 4, 34,  584, 15651,  556696, 24118370, 1213065672, ...];
%e A360337 n=4: [1, 5, 45,  770, 20415,  720141, 31017985, 1553601145, ...];
%e A360337 n=5: [1, 6, 57,  974, 25566,  894462, 38301069, 1910335764, ...];
%e A360337 n=6: [1, 7, 70, 1197, 31129, 1080296, 45986885, 2283964852, ...];
%e A360337 n=7: [1, 8, 84, 1440, 37130, 1278312, 54095632, 2675214000, ...]; ...
%e A360337 Compare to the table of coefficients in (1 + x*A(x)^(3*n+1))^(n+1):
%e A360337 n=0: [1, 1,   1,    7,   124,    3446,   125706,    5540958, ...];
%e A360337 n=1: [1, 2,   9,   76,  1252,   32742,  1150522,   49515052, ...];
%e A360337 n=2: [1, 3,  24,  253,  4179,  103866,  3499510,  146421240, ...];
%e A360337 n=3: [1, 4,  46,  584, 10061,  242520,  7836278,  317454824, ...];
%e A360337 n=4: [1, 5,  75, 1115, 20415,  487566, 15193230,  594390940, ...];
%e A360337 n=5: [1, 6, 111, 1892, 37119,  894462, 27139545, 1025356992, ...];
%e A360337 n=6: [1, 7, 154, 2961, 62412, 1538698, 45986885, 1682957396, ...];
%e A360337 n=7: [1, 8, 204, 4368, 98894, 2519232, 75032832, 2675214000, ...]; ...
%e A360337 to see that the main diagonals of the tables are the same.
%o A360337 (PARI) {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(3*m+1))^(m+1))[m+1] - Vec(Ser(A)^(m+1))[m+1])/(m+1) ); A[n+1]}
%o A360337 for(n=0, 20, print1(a(n), ", "))
%Y A360337 Cf. A360336, A360338, A360237, A360345, A302703.
%K A360337 nonn
%O A360337 0,3
%A A360337 _Paul D. Hanna_, Feb 06 2023