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%I #25 Jun 09 2026 18:31:03
%S 1,2,-3,40,-795,22896,-877625,42472704,-2504024775,175277668960,
%T -14288006295189,1336079557627200,-141604073759908331,
%U 16843101697284562896,-2230002746655974502225,326367650557985432023936,-52485113772229506638866575,9226584510326784965254184640,-1764994657079434895818835078573
%N E.g.f. satisfies A(x) = exp(-x) * A(A(x)).
%H Paul D. Hanna, <a href="/A396799/b396799.txt">Table of n, a(n) for n = 1..256</a>
%F E.g.f. A(x) = Sum_{n>=1} a(n)*x^n/n! satisfies the following formulas.
%F (1.a) A(x) = x * exp( A^(-1)(x) ).
%F (1.b) A(x) = A^2(x) / exp(x).
%F (1.c) A^2(x) = A^3(x) / exp(A(x)).
%F (1.d) A^3(x) = A^4(x) / exp(A^2(x)).
%F (1.e) A^n(x) = A^(n+1)(x) / exp( A^(n-1)(x) ) for all n.
%F (1.f) A^n(x) = A^(n-1)(x) * exp( A^(n-2)(x) ) for all n.
%F (2.a) A(x) = x*exp( A^(-2)(x) * exp( A^(-4)(x) * exp( A^(-6)(x) * ... * exp( A^(-2*n)(x) * exp( ... ))))), an infinite power tower.
%F (2.b) A^2(x) = A(x) * exp( A^(-1)(x) * exp( A^(-3)(x) * exp( A^(-5)(x) * ... * exp( A^(-2*n+1)(x) * exp( ... ))))), an infinite power tower.
%F (3.a) A^(-1)(x) = log( A(x)/x ).
%F (3.b) x = log( A^2(x)/A(x) ).
%F (3.c) A(x) = log( A^3(x)/A^2(x) ).
%F (3.d) A^2(x) = log( A^4(x)/A^3(x) ).
%F (3.e) A^n(x) = log( A^(n+2)(x) / A^(n+1)(x) ) for all n.
%F (4.a) A(x) = A^3(x) / exp(x + A(x)).
%F (4.b) A(x) = A^4(x) / exp(x + A(x) + A^2(x)).
%F (4.c) A(x) = A^5(x) / exp(x + A(x) + A^2(x) + A^3(x)).
%F (4.d) A(x) = A^n(x) / exp( Sum_{k=0..n-2} A^k(x) ) for n >= 2.
%F (4.e) A^n(x) = A^m(x) * exp( Sum_{k=m-1..n-2} A^k(x) ) for n >= m+1.
%F a(n) = n*A087961(n-1) for n >= 1.
%e E.g.f.: A(x) = x + 2*x^2/2! - 3*x^3/3! + 40*x^4/4! - 795*x^5/5! + 22896*x^6/6! - 877625*x^7/7! + 42472704*x^8/8! + ...
%e where A(x) = exp(-x) * A^2(x).
%e RELATED SERIES.
%e The series reversion of e.g.f. A(x) begins
%e A^(-1)(x) = x - 2*x^2/2! + 15*x^3/3! - 220*x^4/4! + 5025*x^5/5! - 159606*x^6/6! + 6593041*x^7/7! + ... + (-1)^(n-1)*A140054(n)*x^n + ...
%e where A^(-1)(x) = log( A(x)/x ).
%e The second iteration of A(x) begins
%e A^2(x) = x + 4*x^2/2! + 6*x^3/3! + 44*x^4/4! - 600*x^5/5! + 18702*x^6/6! - 732704*x^7/7! + 36050968*x^8/8! + ...
%e where A^2(x) = exp(x) * A(x).
%e The third iteration of A(x) begins
%e A^3(x) = x + 6*x^2/2! + 27*x^3/3! + 156*x^4/4! + 225*x^5/5! + 17118*x^6/6! - 533547*x^7/7! + 27717784*x^8/8! + ...
%e where A^3(x) = exp(x + A(x)) * A(x).
%e The fourth iteration of A(x) begins
%e A^4(x) = x + 8*x^2/2! + 60*x^3/3! + 520*x^4/4! + 4200*x^5/5! + 54324*x^6/6! + 26656*x^7/7! + 26761984*x^8/8! + ...
%e where A^4(x) = exp(x + A(x) + A^2(x)) * A(x).
%o (PARI) \\ from A(x) = x * exp( A^(-1)(x) ).
%o {a(n) = my(A=x); for(k=1,n, A = truncate(A) + x*O(x^k);
%o A = x * exp( serreverse(A) ) +x*O(x^n);); EGF=A; n!*polcoef(A,n)}
%o {upto(n) = a(n); Vec(serlaplace(EGF))}
%o upto(25)
%o (Python)
%o from math import comb
%o def a_list(n: int) -> list[int]:
%o T = [[0] * (n + 1) for _ in range(n + 1)]
%o a = [0] * (n + 1); a[0] = 1
%o for m in range(0, n + 1):
%o g = [0] + [j * a[j - 1] * comb(m - 1, j - 1) for j in range(1, m + 1)]
%o T[m][1] = g[m]
%o for k in range(2, m + 1):
%o T[m][k] = sum(g[j] * T[m - j][k - 1] for j in range(1, m - k + 2))
%o a[m] = 1 - sum(a[k] * T[m][k] for k in range(1, m))
%o return [T[m][1] for m in range(1, n + 1)]
%o print(a_list(20)) # _Peter Luschny_, Jun 09 2026
%Y Cf. A140054, A087962, A087961.
%K sign
%O 1,2
%A _Paul D. Hanna_, Jun 08 2026