A377535 First term of n-th differences of the sequence x^(x-1) for x >= 1.

1, 1, 6, 42, 416, 5210, 79212, 1417094, 29168624, 679100562, 17645739500, 506235093782, 15893604725352, 542039221415354, 19954673671286564, 788708093950072830, 33312472504166976992, 1497371019734704549538, 71368260385615670087388, 3595248209512068272420582, 190872048208819769608101080

Offset: 0

Author

Harri Aaltonen, Oct 31 2024

Keywords

• nonn

Comments

Inverse binomial transform of A000169.

It appears that a(n) is the number of partial functions f on [n] such that every point in [n] is either in the domain of f or in the image of f. Cf. A377763. - Geoffrey Critzer, Nov 06 2024

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..386

Crossrefs

Cf. A000169, A038051.

Programs

Maple

a:= n-> add((j+1)^j*(-1)^(n-j)*binomial(n,j), j=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 31 2024

Mathematica

With[{t = Table[n^(n - 1), {n, 1, 21}]}, Table[Sum[(-1)^(i - j) * Binomial[i, j] * t[[j + 1]], {j, 0, i}], {i, 0, Length[t] - 1}]] (* Amiram Eldar, Oct 31 2024 *)

PARI

lista(nn) = my(v = vector(nn+1, n, n^(n-1)), vv=vector(nn+1)); vv[1] = v[1]; for (n=1, nn, my(w = vector(#v-1, k, v[k+1] - v[k])); vv[n+1] = w[1]; v = w;); vv; \\ Michel Marcus, Oct 31 2024

Formula

G.f.: Sum_{j>=1} A000169(j)*x^(j-1)/(1+x)^j. - Alois P. Heinz, Oct 31 2024