A394968 G.f. A(x) satisfies [x^n] exp( n*x/(1-n*x) - n*A(x) ) = 0 for n >= 1.

1, 2, 9, 72, 865, 13803, 271789, 6318840, 168569145, 5058300495, 168251893645, 6134532696534, 243013370253329, 10384740636942483, 475886186077784085, 23269228564681306208, 1208855249378669543785, 66476561813824749886071, 3857014388643231030003397

Offset

1,2

Links

Seiichi Manyama, Table of n, a(n) for n = 1..382

Formula

a(n) = n^(n-1) + (1/n) * Sum_{k=1..n-1} k * c_n(k) * e_n(n-k),

where c_n(k) = n^(k-1) - a(k) for 1 <= k <= n-1,

and e_n(0) = 1, e_n(k) = (n/k) * Sum_{j=1..k} j * c_n(j) * e_n(k-j) for 1 <= k <= n-1.

Example

The table of coefficients of x^k in exp( n*x/(1-n*x) - n*A(x) ) begins:
n\k | 0  1   2    3        4      5          6
----+----------------------------------------------
  1 | 1, 0, -1,  -8,  -141/2,  -856,  -82195/6, ...
  2 | 1, 0,  0, -10,  -128,   -1698,  -27492,   ...
  3 | 1, 0,  3,   0,  -261/2, -2352,  -82161/2, ...
  4 | 1, 0,  8,  28,     0,   -2212, -152684/3, ...
  5 | 1, 0, 15,  80,   755/2,     0,  -91305/2, ...
  6 | 1, 0, 24, 162,  1152,    6474,       0,   ...
  7 | 1, 0, 35, 280,  5019/2, 20552,  802613/6, ...
  8 | 1, 0, 48, 440,  4672,   46968,  435912,   ...
  9 | 1, 0, 63, 648, 15795/2, 92088, 2062719/2, ...

Prog

(Ruby)
def A394968(n)
  a = [0]
  (1..n).each{|i|
    c = [0] + (1..i - 1).map{|k| i ** (k - 1) - a[k]}
    e = [1]
    (1..i - 1).each{|k| e << i / k.to_r * (1..k).inject(0){|s, j| s + j * c[j] * e[k - j]}}
    a << i ** (i - 1) + (1..i - 1).inject(0){|s, k| s + k * c[k] * e[i - k]}.to_i / i
  }
  a[1..-1]
end
p A394968(20)

Crossrefs

Cf. A317344, A394969.

Sequence in context: A240956 A038035 A133984 * A208898 A108995 A184358

Adjacent sequences: A394965 A394966 A394967 * A394969 A394970 A394971

Keyword

nonn

Author

Seiichi Manyama, Apr 08 2026

Status

approved