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A360345 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(2*n+1))^(n+1) for n >= 0.

%I A360345 #7 Feb 06 2023 04:17:07
%S A360345 1,1,5,62,1214,31269,973485,34993597,1412846469,62926155294,
%T A360345 3053566438307,160005640085764,8992869671470675,539298198547460797,
%U A360345 34364052537634696986,2318526571023659653665,165143229278977841236029,12385688813185721332861730,975844100444710104444582984
%N A360345 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(2*n+1))^(n+1) for n >= 0.
%H A360345 Paul D. Hanna, <a href="/A360345/b360345.txt">Table of n, a(n) for n = 0..300</a>
%F A360345 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
%F A360345 (1) [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(2*n+1))^(n+1) for n>=0.
%F A360345 (2) A(x) = Sum_{n>=0} b(n) * x^n/A(x)^n, where b(n) = [x^n] (1 + x*A(x)^(2*n+1))^(n+1) / (n+1).
%F A360345 a(n) ~ c * d^n * n! * n^alpha, where d = 3.93464558322824528799..., alpha = 1.635402029299..., c = 0.0308525091280143... - _Vaclav Kotesovec_, Feb 06 2023
%e A360345 G.f.: A(x) = 1 + x + 5*x^2 + 62*x^3 + 1214*x^4 + 31269*x^5 + 973485*x^6 + 34993597*x^7 + 1412846469*x^8 + 62926155294*x^9 + ...
%e A360345 RELATED SERIES.
%e A360345 G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
%e A360345 B(x) = 1 + x + 6*x^2 + 78*x^3 + 1543*x^4 + 39810*x^5 + 1239252*x^6 + 44537587*x^7 + 1798314384*x^8 + ... + b(n)*x^n + ...
%e A360345 such that b(n) = [x^n] (1 + x*A(x)^(2*n+1))^(n+1) / (n+1),
%e A360345 as well as b(n) = [x^n] A(x)^(n+1) / (n+1),
%e A360345 so that b(n) begin:
%e A360345 [1/1, 2/2, 18/3, 312/4, 7715/5, 238860/6, 8674764/7, 356300696/8, ...].
%e A360345 ILLUSTRATION OF DEFINITION.
%e A360345 The table of coefficients of x^k in A(x)^(n+1) begins:
%e A360345 n=0: [1, 1,  5,  62,  1214,  31269,   973485,  34993597, ...];
%e A360345 n=1: [1, 2, 11, 134,  2577,  65586,  2025492,  72397390, ...];
%e A360345 n=2: [1, 3, 18, 217,  4104, 103212,  3161648, 112357788, ...];
%e A360345 n=3: [1, 4, 26, 312,  5811, 144428,  4387978, 155030276, ...];
%e A360345 n=4: [1, 5, 35, 420,  7715, 189536,  5710930, 200579975, ...];
%e A360345 n=5: [1, 6, 45, 542,  9834, 238860,  7137401, 249182232, ...];
%e A360345 n=6: [1, 7, 56, 679, 12187, 292747,  8674764, 301023241, ...];
%e A360345 n=7: [1, 8, 68, 832, 14794, 351568, 10330896, 356300696, ...]; ...
%e A360345 Compare to the table of coefficients in (1 + x*A(x)^(2*n+1))^(n+1):
%e A360345 n=0: [1, 1,   1,    5,    62,   1214,    31269,    973485, ...];
%e A360345 n=1: [1, 2,   7,   42,   479,   8750,   216258,   6562156, ...];
%e A360345 n=2: [1, 3,  18,  136,  1560,  26895,   633608,  18631701, ...];
%e A360345 n=3: [1, 4,  34,  312,  3767,  62888,  1412530,  40031684, ...];
%e A360345 n=4: [1, 5,  55,  595,  7715, 128041,  2763270,  75234930, ...];
%e A360345 n=5: [1, 6,  81, 1010, 14172, 238860,  5016947, 131313798, ...];
%e A360345 n=6: [1, 7, 112, 1582, 24059, 418166,  8674764, 219340759, ...];
%e A360345 n=7: [1, 8, 148, 2336, 38450, 696216, 14466592, 356300696, ...]; ...
%e A360345 to see that the main diagonals of the tables are the same.
%o A360345 (PARI) {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(2*m+1))^(m+1))[m+1] - Vec(Ser(A)^(m+1))[m+1])/(m+1) ); A[n+1]}
%o A360345 for(n=0, 20, print1(a(n), ", "))
%Y A360345 Cf. A360342, A360343, A360344, A360346, A360347.
%Y A360345 Cf. A302702, A302703.
%K A360345 nonn
%O A360345 0,3
%A A360345 _Paul D. Hanna_, Feb 05 2023