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
Keywords
Examples
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)))))
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
Crossrefs
Programs
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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
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Mathematica
Table[n! SeriesCoefficient[Nest[Function[x, -Log[1 - x]], x, n], {x, 0, n}], {n, 17}] -
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
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