A087961 E.g.f. f(x) satisfies f(x*f(x)) = exp(x).

1, 1, -1, 10, -159, 3816, -125375, 5309088, -278224975, 17527766896, -1298909663199, 111339963135600, -10892621058454487, 1203078692663183064, -148666849777064966815, 20397978159874089501496, -3087359633660559214050975, 512588028351488053625232480

Offset

0,4

Links

Peter Luschny, Table of n, a(n) for n = 0..282.

Formula

From Seiichi Manyama, Jun 08 2026: (Start)

Let a(n,k) = n! * [x^n] f(x)^k.

a(n,k) = k^n - Sum_{j=1..n-1} binomial(n,j) * a(j,k) * a(n-j,j). (End)

Example

E.g.f.: f(x) = Sum_{n>=0} a(n)*x^n/n! = 1 +1x -1x^2/2! +10x^3/3! -159x^4/4! +3816x^5/5! -125375x^6/6! +-... satisfies f(x*f(x)) = exp(x).
.
The computation following the first formula starts (see Python function):
  [0]       1;
  [1]       1,     1;
  [2]      -1,     2,     1;
  [3]      10,    -3,     6,   1;
  [4]    -159,    40,     0,  12,   1;
  [5]    3816,  -795,   140,  30,  20,  1;
  [6] -125375, 22896, -3480, 360, 120, 30, 1;

Mathematica

n = 18;
f[x_] := Sum[a[i]x^i/i!, {i, 0, n-1}];
Solve[CoefficientList[f[x f[x]] - Exp[x] + O[x]^n, x] == ConstantArray[0, n], a/@Range[0, n-1]][[1,;; , 2]] (* Andrei Zabolotskii, Jun 08 2026 *)

Prog

(SageMath)
from sage.all import QQ, ZZ, PowerSeriesRing, factorial
def aList(n: int) -> list[int]:
    S = PowerSeriesRing(QQ, 'x', default_prec=n + 1)
    x = S.gen()
    exp_x = x.exp()
    coeffs = []
    for k in range(n):
        def coeff_at_k(trial):
            f = S([(coeffs + [trial] + [0]*(n-len(coeffs)-1))[i] / factorial(i)
                  for i in range(n)], prec=n + 1)
            g = x * f
            gp = S(1)
            ck = 0
            for i in range(k + 1):
                ck += f[i] * gp[k]
                gp = (gp * g).truncate(k + 1)
            return ck
        c0 = coeff_at_k(0)
        c1 = coeff_at_k(1)
        coeffs.append((exp_x[k] - c0) / (c1 - c0))
    return [ZZ(c) for c in coeffs]
print(aList(18))  # Peter Luschny, Jun 09 2026
(Python)
from math import comb
def a_list(n: int) -> list[int]:
    T = [[0] * (n + 1) for _ in range(n + 1)]
    a = [0] * n
    a[0] = 1
    for m in range(1, n):
        g = [0] + [j * a[j - 1] for j in range(1, m + 1)]
        T[m][1] = g[m]
        for k in range(2, m + 1):
            s = 0
            for j in range(1, m - k + 2):
                s += comb(m - 1, j - 1) * g[j] * T[m - j][k - 1]
            T[m][k] = s
        s_a = 0
        for k in range(1, m):
            s_a += a[k] * T[m][k]
        a[m] = 1 - s_a
    return a
a_list(283)  # Peter Luschny, Jun 09 2026

Crossrefs

Column k=1 of A396860.

Cf. A087962, A140054, A144681, A184011.

Sequence in context: A245916 A245914 A239762 * A360935 A116041 A284110

Adjacent sequences: A087958 A087959 A087960 * A087962 A087963 A087964

Keyword

sign,changed

Author

Paul D. Hanna, Sep 18 2003

Extensions

a(17)-a(18) from Andrei Zabolotskii, Jun 08 2026

Status

approved