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%I #17 Oct 13 2020 12:36:04
%S 1,1,5,175,18385,3759701,1258735981,630063839035,445962163492385,
%T 429694421369414185,547875295770399220981,903754519692129905068391,
%U 1892423689107542226463430065,4948056864672913520114055888445,15922007799835205487157437619131485,62245856465769048392433555378169339891,292266373167286246870149657443033728860481
%N E.g.f. A(x) satisfies: [x^n] A(x)^(n^2) = n^2 * [x^(n-1)] A(x)^(n^2) for n>=1.
%C Compare e.g.f. to: [x^n] exp(x)^(n^2) = n * [x^(n-1)] exp(x)^(n^2) for n>=1.
%H Paul D. Hanna, <a href="/A300590/b300590.txt">Table of n, a(n) for n = 0..200</a>
%F E.g.f. A(x) satisfies: log(A(x)) = Sum_{n>=1} A300591(n)*x^n, a power series in x with integer coefficients.
%F a(n) ~ c * n!^3 * n^2, where c = 0.1354708370957778563796... - _Vaclav Kotesovec_, Oct 13 2020
%e E.g.f.: A(x) = 1 + x + 5*x^2/2! + 175*x^3/3! + 18385*x^4/4! + 3759701*x^5/5! + 1258735981*x^6/6! + 630063839035*x^7/7! + 445962163492385*x^8/8! + 429694421369414185*x^9/9! + 547875295770399220981*x^10/10! + ...
%e ILLUSTRATION OF DEFINITION.
%e The table of coefficients of x^k in A(x)^(n^2) begins:
%e n=1: [(1), (1), 5/2, 175/6, 18385/24, 3759701/120, 1258735981/720, ...];
%e n=2: [1, (4), (16), 452/3, 10448/3, 2037388/15, 333368656/45, ...];
%e n=3: [1, 9, (117/2), (1053/2), 79803/8, 14107743/40, 1472857749/80, ...];
%e n=4: [1, 16, 160, (4880/3), (78080/3), 11770672/15, 1707161056/45, ...];
%e n=5: [1, 25, 725/2, 27175/6, (1642225/24), (41055625/24), ...];
%e n=6: [1, 36, 720, 11340, 180720, (19548324/5), (703739664/5), ...];
%e n=7: [1, 49, 2597/2, 154399/6, 11125009/24, (1138996229/120), (205943018701/720), ...]; ...
%e in which the coefficients in parenthesis are related by
%e 1 = 1*(1); 16 = 2^2*(4); 1053/2 = 3^2*(117/2); 78080/3 = 4^2*(4880/3); 41055625/24 = 5^2*(1642225/24); ...
%e illustrating that: [x^n] A(x)^(n^2) = n^2 * [x^(n-1)] A(x)^(n^2).
%e LOGARITHMIC PROPERTY.
%e The logarithm of the e.g.f. is the integer series:
%e log(A(x)) = x + 2*x^2 + 27*x^3 + 736*x^4 + 30525*x^5 + 1715454*x^6 + 123198985*x^7 + 10931897664*x^8 + 1172808994833*x^9 + 149774206572050*x^10 + ... + A300591(n)*x^n + ...
%o (PARI) {a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^((#A-1)^2)); A[#A] = ((#A-1)^2*V[#A-1] - V[#A])/(#A-1)^2 ); n!*A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A182962, A300870, A296170, A300591, A300592, A300594.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Mar 09 2018
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