A394982 - OEIS

A394982

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

%I #21 Apr 12 2026 08:39:58

%S 1,2,9,8,240,2379,13860,379696,4839048,63662440,1689057480,

%T 27130512294,542163301680,14594796519968,290672365591620,

%U 7336896240946048,206683456014205872,4951125201195562248,144532076753308070352,4337041161081941767680,120922365106904865110208

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

%H Seiichi Manyama, <a href="/A394982/b394982.txt">Table of n, a(n) for n = 1..477</a>

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

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

%F c_n(k) = b_n(k) - a(k) for 1 <= k <= n-1,

%F 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.

%e The table of coefficients of x^k in exp( n*x*(1+n*x+(n*x)^2) - n*A(x) ) begins:

%e n\k | 0 1 2 3 4 5 6

%e ----+--------------------------------------------

%e 1 | 1, 0, -1, -8, -15/2, -232, -14035/6, ...

%e 2 | 1, 0, 0, -10, -16, -480, -4708, ...

%e 3 | 1, 0, 3, 0, -39/2, -720, -14409/2, ...

%e 4 | 1, 0, 8, 28, 0, -736, -27884/3, ...

%e 5 | 1, 0, 15, 80, 145/2, 0, -17465/2, ...

%e 6 | 1, 0, 24, 162, 240, 2448, 0, ...

%e 7 | 1, 0, 35, 280, 1113/2, 8120, 166397/6, ...

%e 8 | 1, 0, 48, 440, 1088, 19200, 93128, ...

%e 9 | 1, 0, 63, 648, 3825/2, 38664, 451359/2, ...

%o (Ruby)

%o def A394982(n)

%o a = [0]

%o (1..n).each{|i|

%o b = [0] + (1..i).map{|k| k > 3 ? 0 : i ** (k - 1)}

%o c = [0] + (1..i - 1).map{|k| b[k] - a[k]}

%o e = [1]

%o (1..i - 1).each{|k| e << i / k.to_r * (1..k).inject(0){|s, j| s + j * c[j] * e[k - j]}}

%o a << b[i] + (1..i - 1).inject(0){|s, k| s + k * c[k] * e[i - k]}.to_i / i

%o }

%o a[1..-1]

%o end

%o p A394982(21)

%Y Cf. A317344, A394968.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Apr 08 2026