A370017 Expansion of g.f. exp( Sum_{n>=1} ( Sum_{k>=1} k^n*x^k )^n * (1-x)^n / n ).

1, 1, 2, 6, 24, 134, 1054, 11848, 188498, 4229252, 132827660, 5831280558, 357547362450, 30623840955096, 3671208716930842, 616066177338250188, 145118327950242179484, 47979462271120402757058, 22322388348068543767280728, 14614554870662196578923073494, 13488493387943242211496467931272

Offset: 0

Author

Paul D. Hanna, Feb 14 2024

Keywords

• nonn

Comments

a(n)^(1/n^2) tends to 2^(1/4). - Vaclav Kotesovec, Feb 19 2024

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 24*x^4 + 134*x^5 + 1054*x^6 + 11848*x^7 + 188498*x^8 + 4229252*x^9 + 132827660*x^10 + 5831280558*x^11 + 357547362450*x^12 +...
log(A(x)) = x + 3*x^2/2 + 13*x^3/3 + 71*x^4/4 + 531*x^5/5 + 5367*x^6/6 + 74313*x^7/7 + 1401295*x^8/8 + 36221143*x^9/9 + 1283583423*x^10/10 + ...
The logarithm of g.f. A(x) equals the series:
log(A(x)) = Sum_{n>=1} (x + 2^n*x^2 + 3^n*x^3 +...+ k^n*x^k +...)^n * (1-x)^n/n,
or,
log(A(x)) = (x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 +...)*(1-x) +
(x + 2^2*x^2 + 3^2*x^3 + 4^2*x^4 + 5^2*x^5 +...)^2 * (1-x)^2/2 +
(x + 2^3*x^2 + 3^3*x^3 + 4^3*x^4 + 5^3*x^5 +...)^3 * (1-x)^3/3 +
(x + 2^4*x^2 + 3^4*x^3 + 4^4*x^4 + 5^4*x^5 +...)^4 * (1-x)^4/4 + ...
This logarithmic series can be written using the Eulerian numbers like so:
log(A(x)) = x/(1-x) + (x + x^2)^2/(1-x)^4/2 + (x + 4*x^2 + x^3)^3/(1-x)^9/3 + (x + 11*x^2 + 11*x^3 + x^4)^4/(1-x)^16/4 + (x + 26*x^2 + 66*x^3 + 26*x^4 + x^5)^5/(1-x)^25/5 + (x + 57*x^2 + 302*x^3 + 302*x^4 + 57*x^5 + x^6)^6/(1-x)^36/6 + ...
+ [ Sum_{k=1..n} A008292(n,k) * x^k ]^n / (1-x)^(n^2)/n +...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..120

Crossrefs

Cf. A156170, A008292.

Programs

PARI

{A008292(n, k) = sum(j=0, k, (-1)^j * (k-j)^n * binomial(n+1, j))}
{a(n) = my(A=1, Oxn=x*O(x^n)); A = exp( sum(m=1, n+1, sum(k=1, m, A008292(m, k)*x^k +Oxn)^m / (1-x +Oxn)^(m^2) / m ) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

(1) exp( Sum_{n>=1} ( Sum_{k>=1} k^n*x^k )^n * (1-x)^n / n ).

(2) exp( Sum_{n>=1} ( Sum_{k=1..n} A008292(n,k) * x^k )^n * (1-x)^(-n^2) / n ), where A008292 are the Eulerian numbers.