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%I #8 Feb 06 2023 12:08:43
%S 1,1,8,152,4452,169952,7807014,413004366,24498135084,1601156353073,
%T 113923669100054,8747479687135221,720094655642863843,
%U 63228142773931718867,5897275794731167406208,582262196337324537825772,60678076577289308772410092,6656827638797910274281675184
%N G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(3*n+2))^(n+1) for n >= 0.
%H Paul D. Hanna, <a href="/A360338/b360338.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)^(3*n+2))^(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)^(3*n+2))^(n+1) / (n+1).
%F a(n) ~ c * d^n * n! * n^alpha, where d = 5.7189630165873859806..., alpha = 1.7741677239019..., c = 0.0189263625965882... - _Vaclav Kotesovec_, Feb 06 2023
%e G.f.: A(x) = 1 + x + 8*x^2 + 152*x^3 + 4452*x^4 + 169952*x^5 + 7807014*x^6 + 413004366*x^7 + 24498135084*x^8 + 1601156353073*x^9 + ...
%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 + 9*x^2 + 177*x^3 + 5237*x^4 + 200533*x^5 + 9220635*x^6 + 487973429*x^7 + 28953420029*x^8 + ... + b(n)*x^n + ...
%e such that b(n) = [x^n] (1 + x*A(x)^(3*n+2))^(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 [1/1, 2/2, 27/3, 708/4, 26185/5, 1203198/6, 64544445/7, 3903787432/8, ...].
%e ILLUSTRATION OF DEFINITION.
%e The table of coefficients of x^k in A(x)^(n+1) begins:
%e n=0: [1, 1, 8, 152, 4452, 169952, 7807014, 413004366, ...];
%e n=1: [1, 2, 17, 320, 9272, 351240, 16048268, 845695400, ...];
%e n=2: [1, 3, 27, 505, 14484, 544512, 24744926, 1298895150, ...];
%e n=3: [1, 4, 38, 708, 20113, 750448, 33919144, 1773460112, ...];
%e n=4: [1, 5, 50, 930, 26185, 969761, 43594110, 2270282630, ...];
%e n=5: [1, 6, 63, 1172, 32727, 1203198, 53794085, 2790292344, ...];
%e n=6: [1, 7, 77, 1435, 39767, 1451541, 64544445, 3334457687, ...];
%e n=7: [1, 8, 92, 1720, 47334, 1715608, 75871724, 3903787432, ...]; ...
%e Compare to the table of coefficients in (1 + x*A(x)^(3*n+2))^(n+1):
%e n=0: [1, 1, 2, 17, 320, 9272, 351240, 16048268, ...];
%e n=1: [1, 2, 11, 110, 1985, 54730, 2003692, 89482592, ...];
%e n=2: [1, 3, 27, 325, 5928, 157206, 5548868, 241397910, ...];
%e n=3: [1, 4, 50, 708, 13443, 348700, 11883916, 502177632, ...];
%e n=4: [1, 5, 80, 1305, 26185, 675816, 22359050, 916389110, ...];
%e n=5: [1, 6, 117, 2162, 46170, 1203198, 38962709, 1549794426, ...];
%e n=6: [1, 7, 161, 3325, 75775, 2016966, 64544445, 2498939864, ...];
%e n=7: [1, 8, 212, 4840, 117738, 3228152, 103075540, 3903787432, ...]; ...
%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)^(3*m+2))^(m+1))[m+1] - Vec(Ser(A)^(m+1))[m+1])/(m+1) ); A[n+1]}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A360336, A360337, A360234, A360346.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Feb 06 2023
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