A305133 E.g.f.: (1-x) / (exp(-x) - x).

1, 1, 3, 16, 113, 996, 10537, 130054, 1834513, 29111896, 513307601, 9955832514, 210652214665, 4828548335092, 119193293536969, 3152465052989326, 88935973854834593, 2665836978234855984, 84608363388300429601, 2834484567764492239354, 99956558270008377397081, 3701159405682998540166796, 143571313108884280622221913, 5822409005523822986360056326

Offset: 0

Author

Paul D. Hanna, Jun 15 2018

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 113*x^4/4! + 996*x^5/5! + 10537*x^6/6! + 130054*x^7/7! + 1834513*x^8/8! + 29111896*x^9/9! + ...
RELATED TABLE.
The table of coefficients of x^k in exp(n*x) * A(x) begins:
n=0: [1, (1), 3/2, 8/3, 113/24, 83/10, 10537/720, 65027/2520, ...];
n=1: [(1), 2, (3), 29/6, 25/3, 1757/120, 929/36, 45863/1008, ...];
n=2: [1, (3), 11/2, (9), 361/24, 1559/60, 729/16, 101107/1260, ...];
n=3: [1, 4, (9), 97/6, (82/3), 1863/40, 3637/45, 714319/5040, ...];
n=4: [1, 5, 27/2, (82/3), 1169/24, (251/3), 103801/720, 632897/2520, ...];
n=5: [1, 6, 19, 87/2, (251/3), 17821/120, (5147/20), 2250499/5040, ...];
n=6: [1, 7, 51/2, 197/3, 3305/24, (5147/20), 65633/144, (14293/18), ...];
n=7: [1, 8, 33, 569/6, 652/3, 51893/120, (14293/18), 7078303/5040, ...]; ...
in which terms along the diagonals (enclosed in parenthesis) are equal:
[x^n] exp((n+1)*x) * A(x) = [x^(n+1)] exp(n*x) * A(x) for n >= 0.
		

Crossrefs

Cf. A072597, A305990.

Programs

Mathematica

With[{nn=30},CoefficientList[Series[(1-x)/(Exp[-x]-x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 15 2022 *)

PARI

{a(n) = n!*polcoeff( (1-x) / (exp(-x +x*O(x^n)) - x), n)}
for(n=0,30,print1(a(n),", "))

Formula

E.g.f. A(x) satisfies: [x^n] exp((n+1)*x) * A(x) = [x^(n+1)] exp(n*x) * A(x) for n >= 0.

a(n) ~ n! * (1 - LambertW(1)) / ((1 + LambertW(1)) * LambertW(1)^(n+1)). - Vaclav Kotesovec, Jun 16 2018

a(n) = 1 + n * Sum_{k=1..n-1} binomial(n-1,k) * a(k). - Ilya Gutkovskiy, Aug 08 2020