|
%I #12 Oct 14 2020 02:39:55
%S 1,1,9,1483,976825,1507281021,4409747597401,21744850191313999,
%T 167557834535988306033,1913194223179191462419065,
%U 31110747474489521617502800201,698529144858380953105954686101811,21123268203104470199318422678044241129,842759726425517953579189712209822358428213,43599233739340643789919321494623289001407934105
%N E.g.f. A(x) satisfies: [x^n] A(x)^(n^3) = n^3 * [x^(n-1)] A(x)^(n^3) for n>=1.
%C Compare e.g.f. to: [x^n] exp(x)^(n^3) = n^2 * [x^(n-1)] exp(x)^(n^3) for n>=1.
%H Paul D. Hanna, <a href="/A300594/b300594.txt">Table of n, a(n) for n = 0..200</a>
%F E.g.f. A(x) satisfies: log(A(x)) = Sum_{n>=1} A300595(n)*x^n, a power series in x with integer coefficients.
%F a(n) ~ c * n!^4 * n^3, where c = 0.40774346023... - _Vaclav Kotesovec_, Oct 14 2020
%e E.g.f.: A(x) = 1 + x + 9*x^2/2! + 1483*x^3/3! + 976825*x^4/4! + 1507281021*x^5/5! + 4409747597401*x^6/6! + 21744850191313999*x^7/7! + 167557834535988306033*x^8/8! + 1913194223179191462419065*x^9/9! + 31110747474489521617502800201*x^10/10! + ...
%e ILLUSTRATION OF DEFINITION.
%e The table of coefficients of x^k in A(x)^(n^3) begins:
%e n=1: [(1), (1), 9/2, 1483/6, 976825/24, 502427007/40, 4409747597401/720]
%e n=2: [1, (8), (64), 6856/3, 1022528/3, 1543097816/15, 2237393526784/45]
%e n=3: [1, 27, (945/2), (25515/2), 10692675/8, 14849374869/40, 1397853444500
%e n=4: [1, 64, 2304, (226880/3), (14520320/3), 5124803136/5, 20241220116736/
%e n=5: [1, 125, 16625/2, 2510375/6, (553359625/24), (69169953125/24), ...];
%e n=6: [1, 216, 24192, 1918728, 131302080, (56555402904/5), (12215967027264/5), ...]; ...
%e in which the coefficients in parenthesis are related by
%e 1 = 1*1; 64 = 2^3*8; 25515/2 = 3^3*945/2; 14520320/3 = 4^3*226880/3; ...
%e illustrating that: [x^n] A(x)^(n^3) = n^3 * [x^(n-1)] A(x)^(n^3).
%e LOGARITHMIC PROPERTY.
%e The logarithm of the e.g.f. is the integer series:
%e log(A(x)) = x + 4*x^2 + 243*x^3 + 40448*x^4 + 12519125*x^5 + 6111917748*x^6 + 4308276119854*x^7 + 4151360558858752*x^8 + 5268077625693186225*x^9 + 8567999843251994553500*x^10 + ... + A300595(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)^3)); A[#A] = ((#A-1)^3*V[#A-1] - V[#A])/(#A-1)^3 ); EGF=Ser(A); n!*A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A182962, A296170, A300590, A300592, A300595.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Mar 09 2018
| 