A377535 - OEIS

%I #40 Nov 07 2024 09:51:47

%S 1,1,6,42,416,5210,79212,1417094,29168624,679100562,17645739500,

%T 506235093782,15893604725352,542039221415354,19954673671286564,

%U 788708093950072830,33312472504166976992,1497371019734704549538,71368260385615670087388,3595248209512068272420582,190872048208819769608101080

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

%C Inverse binomial transform of A000169.

%C 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

%H Alois P. Heinz, <a href="/A377535/b377535.txt">Table of n, a(n) for n = 0..386</a>

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

%p a:= n-> add((j+1)^j*(-1)^(n-j)*binomial(n,j), j=0..n):

%p seq(a(n), n=0..20);  # _Alois P. Heinz_, Oct 31 2024

%t 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 *)

%o (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

%Y Cf. A000169, A038051.

%K nonn

%O 0,3

%A _Harri Aaltonen_, Oct 31 2024