A319938 O.g.f. A(x) satisfies [x^n] exp(-n*A(x)) / (1 - n*x) = 0, for n > 0.

1, 1, 3, 18, 165, 2019, 30688, 554784, 11591649, 274313325, 7242994143, 210931834662, 6713206636084, 231754182524900, 8624280230971980, 344124280164153056, 14656294893872323449, 663624782214112471329, 31833832291287920426617, 1612762327644980719082470, 86050799297228500838101677, 4823357354919905244973170883, 283375597845431500054861239512

Offset

1,3

Comments

It is remarkable that this sequence should consist entirely of integers.

Compare to: [x^n] exp(-n*G(x)) * (1 + n*x) = 0, for n > 0, when G(x) = x - x*G(x)*G'(x), where G(-x)/(-x) is the o.g.f. of A088716.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..300

Formula

a(n) ~ c * n^(n-1), where c = 0.335949071234... - Vaclav Kotesovec, Oct 22 2020

From Seiichi Manyama, Apr 10 2026: (Start)

a(n) = n^(n-2) + (1/n) * Sum_{k=1..n-1} k * c_n(k) * e_n(n-k),

where c_n(k) = n^(k-1)/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. (End)

Example

O.g.f.: A(x) = x + x^2 + 3*x^3 + 18*x^4 + 165*x^5 + 2019*x^6 + 30688*x^7 + 554784*x^8 + 11591649*x^9 + 274313325*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(-n*A(x)) / (1 - n*x) begins:
  n=1: [1, 0, -1, -16,  -423,  -19616, -1444625, -154014624, ...];
  n=2: [1, 0,  0, -20,  -768,  -38832, -2895680, -308705280, ...];
  n=3: [1, 0,  3,   0,  -783,  -53568, -4309605, -465802704, ...];
  n=4: [1, 0,  8,  56,     0,  -50144, -5307200, -616050432, ...];
  n=5: [1, 0, 15, 160,  2265,       0, -4729025, -711963600, ...];
  n=6: [1, 0, 24, 324,  6912,  145584,        0, -613885824, ...];
  n=7: [1, 0, 35, 560, 15057,  460768, 13696795,          0, ...];
  n=8: [1, 0, 48, 880, 28032, 1050432, 44437120, 1769051136, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 3*x^2/2! + 25*x^3/3! + 529*x^4/4! + 22581*x^5/5! + 1598011*x^6/6! + 166508413*x^7/7! + 23765885025*x^8/8! + ...
exp(-A(x)) = 1 - x - x^2/2! - 13*x^3/3! - 359*x^4/4! - 17501*x^5/5! - 1326929*x^6/6! - 143902249*x^7/7! - 21072159247*x^8/8! + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(-m*x*Ser(A))/(1-m*x +x^2*O(x^m)))[m+1]/m ); A[n]}
for(n=1, 30, print1(a(n), ", "))
(Ruby)
def A319938(n)
  a = [0]
  (1..n).each{|i|
    c = [0] + (1..i - 1).map{|k| i ** (k - 1) / k.to_r - 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 - 2) + (1..i - 1).inject(0){|s, k| s + k * c[k] * e[i - k]}.to_i / i
  }
  a[1..-1]
end
p A319938(30) # Seiichi Manyama, Apr 10 2026

Crossrefs

Cf. A088716, A321085, A319939, A319940, A320414, A320417.

Sequence in context: A302585 A107403 A385347 * A385501 A377074 A053513

Adjacent sequences: A319935 A319936 A319937 * A319939 A319940 A319941

Keyword

nonn

Author

Paul D. Hanna, Oct 09 2018

Status

approved