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%I #15 Aug 11 2021 17:00:28
%S 1,1,3,43,2041,197721,31094251,7086479443,2187876597873,
%T 874871971357681,438740658523346131,269314248304239932091,
%U 198529013874402868930153,173067121551267519897494473,176154202119865662835343738811,207099741506845262022248534098531,278645958801870115911315221474653921,425605862347493892454320041743878801633
%N E.g.f. A(x) satisfies: [x^n] A(x)^(n*(n+1)) = 2*n * [x^(n-1)] A(x)^(n*(n+1)) for n>=1.
%C Compare to: [x^n] exp(x)^(n*(n+1)) = (n+1) * [x^(n-1)] exp(x)^(n*(n+1)) for n>=1.
%H Paul D. Hanna, <a href="/A300873/b300873.txt">Table of n, a(n) for n = 0..300</a>
%F a(n) ~ c * d^n * n!^2 / n^3, where d = -4/(LambertW(-2*exp(-2))*(2 + LambertW(-2*exp(-2)))) = 6.17655460948348035823168... and c = 0.75891265... - _Vaclav Kotesovec_, Aug 11 2021
%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 43*x^3/3! + 2041*x^4/4! + 197721*x^5/5! + 31094251*x^6/6! + 7086479443*x^7/7! + 2187876597873*x^8/8! + 874871971357681*x^9/9! + ...
%e ILLUSTRATION OF DEFINITION.
%e The table of coefficients of x^k in A(x)^(n*(n+1)) begins:
%e n=1: [(1), (2), 4, 52/3, 560/3, 52304/15, 4048864/45, 914958416/315, ...];
%e n=2: [1, (6), (24), 108, 864, 67104/5, 1601424/5, 348254352/35, ...];
%e n=3: [1, 12, (84), (504), 3600, 211968/5, 4273776/5, 860107104/35, ...];
%e n=4: [1, 20, 220, (5560/3), (44480/3), 438400/3, 20480720/9, 3534944800/63, ...];
%e n=5: [1, 30, 480, 5580, (55440), (554400), 6991920, 947466000/7, ...];
%e n=6: [1, 42, 924, 14364, 181440, (10403568/5), (124842816/5), 1922103792/5, ...];
%e n=7: [1, 56, 1624, 98224/3, 1566992/3, 107909312/15, (4208547616/45), (58919666624/45), ...]; ...
%e in which the coefficients in parenthesis are related by
%e 2 = 2*1*(1); 24 = 2*2*(6); 504 = 2*3*(84); 44480/3 = 2*4*(5560/3); 554400 = 2*5*(55440); 124842816/5 = 2*6*(10403568/5); ...
%e illustrating that: [x^n] A(x)^(n*(n+1)) = 2*n * [x^(n-1)] A(x)^(n*(n+1)).
%e LOGARITHMIC PROPERTY.
%e The logarithm of the e.g.f. is the integer series:
%e log(A(x)) = x + x^2 + 6*x^3 + 78*x^4 + 1560*x^5 + 41484*x^6 + 1361640*x^7 + 52824144*x^8 + 2355612192*x^9 + 118455668960*x^10 + ... + A300874(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)*(#A))); A[#A] = (2*(#A-1)*V[#A-1] - V[#A])/(#A-1)/(#A) ); EGF=Ser(A); n!*A[n+1]}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A300874, A300870, A295811, A300590, A296170, A182962.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Mar 14 2018
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