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

1, 2, 15, 234, 6170, 245755, 13761937, 1030431500, 99399019626, 12003835242090, 1773907219147800, 314880916127332489, 66109411013740671200, 16204039283106534720952, 4585484528618722750937783, 1483746673734716952089913364, 544359300175753347889146067840

Offset

1,2

Links

Seiichi Manyama, Table of n, a(n) for n = 1..246

Jekuthiel Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353. [Annotated scanned copy]

N. J. A. Sloane, Transforms

Formula

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

Example

The initial coefficients of successive iterations of e.g.f. A(x) = -log(1 - x) are as follows:
n = 1: 0, (1), 1,   2,    6,    24,  ... e.g.f. A(x)
n = 2: 0,  1, (2),  7,   35,   228,  ... e.g.f. A(A(x))
n = 3: 0,  1,  3, (15), 105,   947,  ... e.g.f. A(A(A(x)))
n = 4: 0,  1,  4,  26, (234), 2696,  ... e.g.f. A(A(A(A(x))))
n = 5: 0,  1,  5,  40,  440, (6170), ... e.g.f. A(A(A(A(A(x)))))

Maple

g:= x-> -log(1-x):
a:= n-> n! * coeff(series((g@@n)(x), x, n+1), x, n):
seq(a(n), n=1..19);  # Alois P. Heinz, Feb 11 2022

Mathematica

Table[n! SeriesCoefficient[Nest[Function[x, -Log[1 - x]], x, n], {x, 0, n}], {n, 17}]

Prog

(PARI) T(n, k) = if(k==1, (n-1)!, sum(j=1, n, abs(stirling(n, j, 1))*T(j, k-1)));
a(n) = T(n, n); \\ Seiichi Manyama, Feb 11 2022

Crossrefs

Cf. A000268, A000310, A000359, A000406, A001765, A003713, A104150, A139383, A158832, A174482, A261280.

Sequence in context: A294043 A326095 A212370 * A192561 A356587 A227098

Adjacent sequences: A302355 A302356 A302357 * A302359 A302360 A302361

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 06 2018

Status

approved