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A360343 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 A360343 #9 Feb 06 2023 04:10:27
%S A360343 1,1,3,31,526,11907,328980,10580531,384937042,15549217485,
%T A360343 688430225102,33096289502982,1715499922758709,95339852384471586,
%U A360343 5655337634718941111,356683962066445400017,23840465113068534382248,1683771696557415075462436,125327912444852044066759399
%N A360343 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 A360343 Paul D. Hanna, <a href="/A360343/b360343.txt">Table of n, a(n) for n = 0..300</a>
%F A360343 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
%F A360343 (1) [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(2*n-1))^(n+1) for n>=0.
%F A360343 (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 A360343 a(n) ~ c * d^n * n! * n^alpha, where d = 3.93464558322824528799..., alpha = 0.5984002265754..., c = 0.08321697608093... - _Vaclav Kotesovec_, Feb 06 2023
%e A360343 G.f.: A(x) = 1 + x + 3*x^2 + 31*x^3 + 526*x^4 + 11907*x^5 + 328980*x^6 + 10580531*x^7 + 384937042*x^8 + 15549217485*x^9 + ...
%e A360343 RELATED SERIES.
%e A360343 G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
%e A360343 B(x) = 1 + x + 4*x^2 + 41*x^3 + 687*x^4 + 15433*x^5 + 424524*x^6 + 13620842*x^7 + 495005025*x^8 + ... + b(n)*x^n + ...
%e A360343 such that b(n) = [x^n] (1 + x*A(x)^(2*n-1))^(n+1) / (n+1),
%e A360343 as well as b(n) = [x^n] A(x)^(n+1) / (n+1),
%e A360343 so that {b(n)} begins:
%e A360343 [1/1, 2/2, 12/3, 164/4, 3435/5, 92598/6, 2971668/7, 108966736/8, ...].
%e A360343 ILLUSTRATION OF DEFINITION.
%e A360343 The table of coefficients of x^k in A(x)^(n+1) begins:
%e A360343   n=0: [1, 1,  3,  31,  526,  11907,  328980,  10580531, ...];
%e A360343   n=1: [1, 2,  7,  68, 1123,  25052,  685891,  21923076, ...];
%e A360343   n=2: [1, 3, 12, 112, 1800,  39555, 1072896,  34076544, ...];
%e A360343   n=3: [1, 4, 18, 164, 2567,  55548, 1492336,  47093172, ...];
%e A360343   n=4: [1, 5, 25, 225, 3435,  73176, 1946745,  61028770, ...];
%e A360343   n=5: [1, 6, 33, 296, 4416,  92598, 2438866,  75942984, ...];
%e A360343   n=6: [1, 7, 42, 378, 5523, 113988, 2971668,  91899578, ...];
%e A360343   n=7: [1, 8, 52, 472, 6770, 137536, 3548364, 108966736, ...]; ...
%e A360343 Compare to the table of coefficients in (1 + x*A(x)^(2*n-1))^(n+1):
%e A360343   n=0: [1, 1,  -1,   -2,   -26,   -463,  -10778,   -303048, ...];
%e A360343   n=1: [1, 2,   3,    8,    69,   1120,   24937,    683012, ...];
%e A360343   n=2: [1, 3,  12,   55,   444,   6351,  132492,   3504654, ...];
%e A360343   n=3: [1, 4,  26,  164,  1411,  18560,  357624,   9024812, ...];
%e A360343   n=4: [1, 5,  45,  360,  3435,  43926,  785715,  18700710, ...];
%e A360343   n=5: [1, 6,  69,  668,  7134,  92598, 1570420,  35086104, ...];
%e A360343   n=6: [1, 7,  98, 1113, 13279, 179816, 2971668,  62645353, ...];
%e A360343   n=7: [1, 8, 132, 1720, 22794, 327032, 5403036, 108966736, ...]; ...
%e A360343 to see that the main diagonals of the tables are the same.
%o A360343 (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 A360343 for(n=0, 20, print1(a(n), ", "))
%Y A360343 Cf. A360342, A360344, A360345, A360346, A360347.
%Y A360343 Cf. A360231, A302702.
%K A360343 nonn
%O A360343 0,3
%A A360343 _Paul D. Hanna_, Feb 05 2023