A300597 O.g.f. A(x) satisfies: [x^n] exp( n^4 * A(x) ) = n^4 * [x^(n-1)] exp( n^4 * A(x) ) for n>=1.

1, 8, 2187, 2351104, 6153518125, 31779658925496, 287364845865893467, 4200677982722915635200, 93566442152660422280250537, 3030525904161802498705606745000, 137355046868929476532154243693393581, 8436685562091750543736612601781557411328, 683522945769518614776208838188411394718328617

Offset: 1

Author

Paul D. Hanna, Mar 09 2018

Keywords

• nonn

Comments

Compare to: [x^n] exp( n^4 * x ) = n^3 * [x^(n-1)] exp( n^4 * x ) for n>=1.

It is remarkable that this sequence should consist entirely of integers.

Examples

			O.g.f.: A(x) = x + 8*x^2 + 2187*x^3 + 2351104*x^4 + 6153518125*x^5 + 31779658925496*x^6 + 287364845865893467*x^7 + 4200677982722915635200*x^8 + ...
where
exp(A(x)) = 1 + x + 17*x^2/2! + 13171*x^3/3! + 56479849*x^4/4! + 738706542221*x^5/5! + 22885801082965201*x^6/6! + 1448479282286023114807*x^7/7! + ... + A300596(n)*x^n/n! + ...
such that: [x^n] exp( n^4 * A(x) ) = n^4 * [x^(n-1)] exp( n^4 * A(x) ).
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A300596, A296171, A300591, A300593, A300595.

Programs

PARI

{a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^((#A-1)^4)); A[#A] = ((#A-1)^4*V[#A-1] - V[#A])/(#A-1)^4 ); polcoeff( log(Ser(A)), n)}
for(n=1, 20, print1(a(n), ", "))

Formula

O.g.f. equals the logarithm of the e.g.f. of A300596.