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A302358 a(n) = coefficient of x^n in the n-th iteration (n-fold self-composition) of e.g.f. -log(1 - x).

%I A302358 #12 Feb 11 2022 09:28:20
%S A302358 1,2,15,234,6170,245755,13761937,1030431500,99399019626,
%T A302358 12003835242090,1773907219147800,314880916127332489,
%U A302358 66109411013740671200,16204039283106534720952,4585484528618722750937783,1483746673734716952089913364,544359300175753347889146067840
%N A302358 a(n) = coefficient of x^n in the n-th iteration (n-fold self-composition) of e.g.f. -log(1 - x).
%H A302358 Seiichi Manyama, <a href="/A302358/b302358.txt">Table of n, a(n) for n = 1..246</a>
%H A302358 Jekuthiel Ginsburg, <a href="/A000405/a000405.pdf">Iterated exponentials</a>, Scripta Math., 11 (1945), 340-353. [Annotated scanned copy]
%H A302358 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A302358 a(n) = T(n,n), T(n,k) = Sum_{j=1..n} |Stirling1(n,j)| * T(j,k-1), k>1, T(n,1) = (n-1)!. - _Seiichi Manyama_, Feb 11 2022
%e A302358 The initial coefficients of successive iterations of e.g.f. A(x) = -log(1 - x) are as follows:
%e A302358 n = 1: 0, (1), 1,   2,    6,    24,  ... e.g.f. A(x)
%e A302358 n = 2: 0,  1, (2),  7,   35,   228,  ... e.g.f. A(A(x))
%e A302358 n = 3: 0,  1,  3, (15), 105,   947,  ... e.g.f. A(A(A(x)))
%e A302358 n = 4: 0,  1,  4,  26, (234), 2696,  ... e.g.f. A(A(A(A(x))))
%e A302358 n = 5: 0,  1,  5,  40,  440, (6170), ... e.g.f. A(A(A(A(A(x)))))
%p A302358 g:= x-> -log(1-x):
%p A302358 a:= n-> n! * coeff(series((g@@n)(x), x, n+1), x, n):
%p A302358 seq(a(n), n=1..19);  # _Alois P. Heinz_, Feb 11 2022
%t A302358 Table[n! SeriesCoefficient[Nest[Function[x, -Log[1 - x]], x, n], {x, 0, n}], {n, 17}]
%o A302358 (PARI) T(n, k) = if(k==1, (n-1)!, sum(j=1, n, abs(stirling(n, j, 1))*T(j, k-1)));
%o A302358 a(n) = T(n, n); \\ _Seiichi Manyama_, Feb 11 2022
%Y A302358 Cf. A000268, A000310, A000359, A000406, A001765, A003713, A104150, A139383, A158832, A174482, A261280.
%K A302358 nonn
%O A302358 1,2
%A A302358 _Ilya Gutkovskiy_, Apr 06 2018