A038125 a(n) = Sum_{k=0..n} (k-n)^k.

1, 1, 0, 0, 1, -1, 0, 6, -19, 29, 48, -524, 2057, -3901, -9632, 129034, -664363, 1837905, 2388688, -67004696, 478198545, -1994889945, 1669470784, 56929813934, -615188040195, 3794477505573, -12028579019536, -50780206473220

Offset: 0

Author

Jim Ferry (jferry(AT)alum.mit.edu)

Keywords

• sign
• easy

Examples

			0^0 = 1,
1^0 - 0^1 = 1,
2^0 - 1^1 + 0^2 = 0,
3^0 - 2^1 + 1^2 - 0^3 = 0,
...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A003101, A026898, A120485.

Programs

Mathematica

Prepend[ Table[ Sum[ (k-n)^k, {k, 0, n} ], {n, 30} ], 1 ]

PARI

my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1+k*x))) \\ Seiichi Manyama, Dec 02 2021

PARI

a(n) = sum(k=0, n, (k-n)^k); \\ Michel Marcus, Dec 03 2021

Formula

G.f.: 1+ sum(k>=0, x^(k+1)/(1+x^(k+1)) ) = 1/Q(0), where Q(k) = 1 - x + x^2*(k+1)/(1 + (k+1)*x/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 10 2014