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%I #8 Dec 08 2017 21:11:54
%S 1,1,-29,-36629,-734559239,-71200423546199,-22459270436075644469,
%T -18407129959728493123679069,-33747438879000326056232288023439,
%U -124162549312926509293620790889452447919,-843670934957017748849439817665935283173590349,-9914324850699841477684471316247032518786477385700389,-191047752973105011101288266443568575709649708408401069796759
%N E.g.f. A(x) satisfies: [x^(n-1)] A(x)^(n^5) = [x^n] A(x)^(n^5) for n>=1.
%C Compare e.g.f. to: [x^(n-1)] exp(x)^n = [x^n] exp(x)^n for n>=1.
%H Paul D. Hanna, <a href="/A296176/b296176.txt">Table of n, a(n) for n = 0..150</a>
%F The logarithm of the e.g.f. A(x) is an integer series:
%F _ log(A(x)) = Sum_{n>=1} A296177(n) * x^n.
%F E.g.f. A(x) satisfies:
%F _ 1/n! * d^n/dx^n A(x)^(n^5) = 1/(n-1)! * d^(n-1)/dx^(n-1) A(x)^(n^5) for n>=1, when evaluated at x = 0.
%e E.g.f.: A(x) = 1 + x - 29*x^2/2! - 36629*x^3/3! - 734559239*x^4/4! - 71200423546199*x^5/5! - 22459270436075644469*x^6/6! - 18407129959728493123679069*x^7/7! - 33747438879000326056232288023439*x^8/8! - 124162549312926509293620790889452447919*x^9/9! - 843670934957017748849439817665935283173590349*x^10/10! +...
%e To illustrate [x^(n-1)] A(x)^(n^5) = [x^n] A(x)^(n^5), form a table of coefficients of x^k in A(x)^(n^5) that begins as
%e n=1: [(1), (1), -29/2, -36629/6, -734559239/24, -71200423546199/120, ...];
%e n=2: [1, (32), (32), -614336/3, -2956631488/3, -285257147669696/15, ...];
%e n=3: [1, 243, (51759/2), (51759/2), -62010059733/8, -5840748850240719/40, ...];
%e n=4: [1, 1024, 508928, (470976512/3), (470976512/3), -9540780758505472/15, ...];
%e n=5: [1, 3125, 9671875/2, 29524484375/6, (86178242265625/24), (86178242265625/24), ...];
%e n=6: [1, 7776, 30116448, 77409815616, 148214160396864, (1099707612312815424/5), (1099707612312815424/5), ...];
%e ...
%e in which the diagonals indicated by parenthesis are equal.
%e Dividing the coefficients of x^(n-1)/(n-1)! in A(x)^(n^5) by n^5, we obtain the following sequence:
%e [1, 1, 213, 919876, 27577037525, 3394159297261776, 1269158820664910885737, 1186717596374463676630699264, ...].
%e LOGARITHMIC PROPERTY.
%e Amazingly, the logarithm of the e.g.f. A(x) is an integer series:
%e log(A(x)) = x - 15*x^2 - 6090*x^3 - 30600650*x^4 - 593306350650*x^5 - 31192838317208826*x^6 - 3652177141294409632400*x^7 - 836986399841753367052602000*x^8 - 342157863774785896821739864893375*x^9 - 232492750600387706453977026534258393375*x^10 +...
%o (PARI) {a(n) = my(A=[1]); for(i=1,n+1, A=concat(A,0); V=Vec(Ser(A)^((#A-1)^5)); A[#A] = (V[#A-1] - V[#A])/(#A-1)^5 ); n!*A[n+1]}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A296177, A296170, A296172, A296174.
%K sign
%O 0,3
%A _Paul D. Hanna_, Dec 07 2017
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