A243954 E.g.f.: A(x) = x/Series_Reversion(x*G(x)) where G(x) = Sum_{n>=0} n^n*x^n/n!.

1, 1, 2, 3, -8, -95, -144, 5299, 51584, -263007, -11283200, -61279229, 2321353728, 49471727617, -217783810048, -25824369466125, -298783278792704, 10591512134358721, 400315590893961216, -536860171776183677, -362306852971970560000, -7292410691309793586719, 227379942363338919903232

Offset: 0

Author

Paul D. Hanna, Jun 20 2014

Keywords

• sign

Comments

Compare to: exp(x) = x/Series_Reversion(x*W(x)) where W(x) = Sum_{n>=0} (n+1)^(n-1)*x^n/n!.

Examples

			G.f.: A(x) = 1 + x + 2*x^2/2! + 3*x^3/3! - 8*x^4/4! - 95*x^5/5! - 144*x^6/6! +...
where
A(x) = 1 + x/A(x) + 4*(x/A(x))^2/2! + 27*(x/A(x))^3/3! + 256*(x/A(x))^4/4! + 3125*(x/A(x))^5/5! +...+ n^n*(x/A(x))^n/n! +...
If we form a table of coefficients of x^k/k! in A(x)^n, like so:
[1, 1,  2,   3,    -8,    -95,    -144,     5299,     51584, ...];
[1, 2,  6,  18,    32,   -150,   -1728,    -1078,    144384, ...];
[1, 3, 12,  51,   192,    375,   -2592,   -29841,     12288, ...];
[1, 4, 20, 108,   568,   2500,    5184,   -48020,   -557056, ...];
[1, 5, 30, 195,  1280,   7845,   38880,    84035,   -983040, ...];
[1, 6, 42, 318,  2472,  18750,  129456,   705894,   1572864, ...];
[1, 7, 56, 483,  4312,  38395,  326592,  2485567,  14680064, ...];
[1, 8, 72, 696,  6992,  70920,  704448,  6588344,  54442368, ...];
[1, 9, 90, 963, 10728, 121545, 1368144, 14890995, 150994944, ...]; ...
then the main diagonal equals (k+1)*k^k for k>=0.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..100

Programs

PARI

{a(n)=n!*polcoeff(x/serreverse(x*sum(m=0, n+1, m^m*x^m/m!)+x^2*O(x^n)), n)}
for(n=0,25,print1(a(n),", "))

Formula

E.g.f. satisfies: A(x) = G(x/A(x)) such that A(x*G(x)) = G(x) = Sum_{n>=0} n^n*x^n/n!.

E.g.f. A(x) satisfies: [x^n/n!] A(x)^(n+1)/(n+1) = n^n.