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A300594 E.g.f. A(x) satisfies: [x^n] A(x)^(n^3) = n^3 * [x^(n-1)] A(x)^(n^3) for n>=1. 8
1, 1, 9, 1483, 976825, 1507281021, 4409747597401, 21744850191313999, 167557834535988306033, 1913194223179191462419065, 31110747474489521617502800201, 698529144858380953105954686101811, 21123268203104470199318422678044241129, 842759726425517953579189712209822358428213, 43599233739340643789919321494623289001407934105 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Compare e.g.f. to: [x^n] exp(x)^(n^3) = n^2 * [x^(n-1)] exp(x)^(n^3) for n>=1.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..200

FORMULA

E.g.f. A(x) satisfies: log(A(x)) = Sum_{n>=1} A300595(n)*x^n, a power series in x with integer coefficients.

a(n) ~ c * n!^4 * n^3, where c = 0.40774346023... - Vaclav Kotesovec, Oct 14 2020

EXAMPLE

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! + ...

ILLUSTRATION OF DEFINITION.

The table of coefficients of x^k in A(x)^(n^3) begins:

n=1: [(1), (1), 9/2, 1483/6, 976825/24, 502427007/40, 4409747597401/720]

n=2: [1, (8), (64), 6856/3, 1022528/3, 1543097816/15, 2237393526784/45]

n=3: [1, 27, (945/2), (25515/2), 10692675/8, 14849374869/40, 1397853444500

n=4: [1, 64, 2304, (226880/3), (14520320/3), 5124803136/5, 20241220116736/

n=5: [1, 125, 16625/2, 2510375/6, (553359625/24), (69169953125/24), ...];

n=6: [1, 216, 24192, 1918728, 131302080, (56555402904/5), (12215967027264/5), ...]; ...

in which the coefficients in parenthesis are related by

1 = 1*1; 64 = 2^3*8; 25515/2 = 3^3*945/2; 14520320/3 = 4^3*226880/3; ...

illustrating that: [x^n] A(x)^(n^3) = n^3 * [x^(n-1)] A(x)^(n^3).

LOGARITHMIC PROPERTY.

The logarithm of the e.g.f. is the integer series:

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 + ...

PROG

(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]}

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

CROSSREFS

Cf. A182962, A296170, A300590, A300592, A300595.

Sequence in context: A230172 A231191 A217267 * A203649 A187131 A167774

Adjacent sequences:  A300591 A300592 A300593 * A300595 A300596 A300597

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Mar 09 2018

STATUS

approved

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Last modified June 15 11:06 EDT 2021. Contains 345048 sequences. (Running on oeis4.)