A296176 E.g.f. A(x) satisfies: [x^(n-1)] A(x)^(n^5) = [x^n] A(x)^(n^5) for n>=1.

1, 1, -29, -36629, -734559239, -71200423546199, -22459270436075644469, -18407129959728493123679069, -33747438879000326056232288023439, -124162549312926509293620790889452447919, -843670934957017748849439817665935283173590349, -9914324850699841477684471316247032518786477385700389, -191047752973105011101288266443568575709649708408401069796759

Offset: 0

Author

Paul D. Hanna, Dec 07 2017

Keywords

• sign

Comments

Compare e.g.f. to: [x^(n-1)] exp(x)^n = [x^n] exp(x)^n for n>=1.

Examples

			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! +...
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
n=1: [(1), (1), -29/2, -36629/6, -734559239/24, -71200423546199/120, ...];
n=2: [1, (32), (32), -614336/3, -2956631488/3, -285257147669696/15, ...];
n=3: [1, 243, (51759/2), (51759/2), -62010059733/8, -5840748850240719/40, ...];
n=4: [1, 1024, 508928, (470976512/3), (470976512/3), -9540780758505472/15, ...];
n=5: [1, 3125, 9671875/2, 29524484375/6, (86178242265625/24), (86178242265625/24), ...];
n=6: [1, 7776, 30116448, 77409815616, 148214160396864, (1099707612312815424/5), (1099707612312815424/5), ...];
...
in which the diagonals indicated by parenthesis are equal.
Dividing the coefficients of x^(n-1)/(n-1)! in A(x)^(n^5) by n^5, we obtain the following sequence:
[1, 1, 213, 919876, 27577037525, 3394159297261776, 1269158820664910885737, 1186717596374463676630699264, ...].
LOGARITHMIC PROPERTY.
Amazingly, the logarithm of the e.g.f. A(x) is an integer series:
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 +...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..150

Crossrefs

Cf. A296177, A296170, A296172, A296174.

Programs

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]}
for(n=0,30,print1(a(n),", "))

Formula

The logarithm of the e.g.f. A(x) is an integer series:

log(A(x)) = Sum{n>=1} A296177(n) * x^n.

E.g.f. A(x) satisfies:

_ 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.