A394982 G.f. A(x) satisfies [x^n] exp( n*x*(1+n*x+(n*x)^2) - n*A(x) ) = 0 for n >= 1.

1, 2, 9, 8, 240, 2379, 13860, 379696, 4839048, 63662440, 1689057480, 27130512294, 542163301680, 14594796519968, 290672365591620, 7336896240946048, 206683456014205872, 4951125201195562248, 144532076753308070352, 4337041161081941767680, 120922365106904865110208

Offset

1,2

Links

Seiichi Manyama, Table of n, a(n) for n = 1..477

Formula

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

where b_n(k) = n^(k-1) for 1 <= k <= 3, and b_n(k) = 0 for k > 3,

c_n(k) = b_n(k) - 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*x)^2) - n*A(x) ) begins:
n\k | 0  1   2    3       4      5         6
----+--------------------------------------------
  1 | 1, 0, -1,  -8,  -15/2,  -232, -14035/6, ...
  2 | 1, 0,  0, -10,  -16,    -480,  -4708,   ...
  3 | 1, 0,  3,   0,  -39/2,  -720, -14409/2, ...
  4 | 1, 0,  8,  28,    0,    -736, -27884/3, ...
  5 | 1, 0, 15,  80,  145/2,     0, -17465/2, ...
  6 | 1, 0, 24, 162,  240,    2448,      0,   ...
  7 | 1, 0, 35, 280, 1113/2,  8120, 166397/6, ...
  8 | 1, 0, 48, 440, 1088,   19200,  93128,   ...
  9 | 1, 0, 63, 648, 3825/2, 38664, 451359/2, ...

Prog

(Ruby)
def A394982(n)
  a = [0]
  (1..n).each{|i|
    b = [0] + (1..i).map{|k| k > 3 ? 0 : i ** (k - 1)}
    c = [0] + (1..i - 1).map{|k| b[k] - 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 << b[i] + (1..i - 1).inject(0){|s, k| s + k * c[k] * e[i - k]}.to_i / i
  }
  a[1..-1]
end
p A394982(21)

Crossrefs

Cf. A317344, A394968.

Sequence in context: A092270 A249225 A191351 * A324553 A230283 A121067

Adjacent sequences: A394979 A394980 A394981 * A394983 A394984 A394985

Keyword

nonn

Author

Seiichi Manyama, Apr 08 2026

Status

approved