A295103 -id:A295103 - cp's OEIS frontend

A179230 n-th derivative of x^(x^x) at x=1.

1, 1, 2, 9, 32, 180, 954, 6524, 45016, 360144, 3023640, 27617832, 271481880, 2775329232, 31188079272, 350827041000, 4441125248640, 54110311240512, 765546040603584, 9938498593229568, 156934910753107200, 2128783325724881280, 37775147271084647424

Offset: 0

Henryk Trappmann (bo198214(AT)gmail.com), Jul 03 2010

sign

First term < 0: a(33) = -868875490363254484795699722301440.

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. A005727. Column k=4 of A215703. Column k=3 of A277537.

Cf. A295103.

a:= n-> n!*coeff(series(subs(x=x+1, x^(x^x) ), x, n+1), x, n):
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 20 2012

Table[ D[ x^(x^x), {x, n}] /. x -> 1, {n, 0, 20}] (* Robert G. Wilson v, Jul 12 2010 *)
NestList[ Factor[ D[ #1, x]] &, x^x^x, 20] /. x -> 1 (* Robert G. Wilson v, Aug 10 2010 *)
Range[0, 22]! CoefficientList[ Series[(1 + x)^(1 + x)^(1 + x), {x, 0, 22}], x] (* Robert G. Wilson v, Feb 03 2013 *)

E.g.f.: (x+1)^((x+1)^(x+1)). - Alois P. Heinz, Aug 21 2012

a(21)-a(22) from Alois P. Heinz, Aug 20 2012

A295028 A(n,k) is (1/n) times the n-th derivative of the k-th tetration of x (power tower of order k) x^^k at x=1; square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 2, 0, 1, 1, 3, 8, 2, 0, 1, 1, 3, 14, 36, 9, 0, 1, 1, 3, 14, 72, 159, -6, 0, 1, 1, 3, 14, 96, 489, 932, 118, 0, 1, 1, 3, 14, 96, 729, 3722, 5627, -568, 0, 1, 1, 3, 14, 96, 849, 6842, 33641, 40016, 4716, 0

Offset: 1

Alois P. Heinz, Nov 12 2017

			Square array A(n,k) begins:
  1,   1,    1,     1,     1,      1,      1,      1, ...
  0,   1,    1,     1,     1,      1,      1,      1, ...
  0,   1,    3,     3,     3,      3,      3,      3, ...
  0,   2,    8,    14,    14,     14,     14,     14, ...
  0,   2,   36,    72,    96,     96,     96,     96, ...
  0,   9,  159,   489,   729,    849,    849,    849, ...
  0,  -6,  932,  3722,  6842,   8642,   9362,   9362, ...
  0, 118, 5627, 33641, 71861, 102941, 118061, 123101, ...

Alois P. Heinz, Antidiagonals n = 1..141, flattened
Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Knuth's up-arrow notation
Wikipedia, Tetration

Columns k=1-10 give: A063524, A005168, A295103, A295104, A295105, A295106, A295107, A295108, A295109, A295110.
Main diagonal gives A136461(n-1).
Cf. A277537, A295027.

f:= proc(n) f(n):= `if`(n=0, 1, (x+1)^f(n-1)) end:
A:= (n, k)-> (n-1)!*coeff(series(f(k), x, n+1), x, n):
seq(seq(A(n, 1+d-n), n=1..d), d=1..14);
# second Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1, `if`(k=0, 0,
      -add(binomial(n-1, j)*b(j, k)*add(binomial(n-j, i)*
      (-1)^i*b(n-j-i, k-1)*(i-1)!, i=1..n-j), j=0..n-1)))
    end:
A:= (n, k)-> b(n, min(k, n))/n:
seq(seq(A(n, 1+d-n), n=1..d), d=1..14);

b[n_, k_] := b[n, k] = If[n == 0, 1, If[k == 0, 0, -Sum[Binomial[n - 1, j]*b[j, k]*Sum[Binomial[n - j, i]*(-1)^i*b[n - j - i, k - 1]*(i - 1)!, {i, 1, n - j}], {j, 0, n - 1}]]];
A[n_, k_] := b[n, Min[k, n]]/n;
Table[A[n, 1 + d - n], {d, 1, 14}, {n, 1, d}] // Flatten (* Jean-François Alcover, May 25 2018, translated from 2nd Maple program *)

A(n,k) = 1/n * [(d/dx)^n x^^k]_{x=1}.
A(n,k) = (n-1)! * [x^n] (x+1)^^k.
A(n,k) = Sum_{i=0..min(n,k)} A295027(n,i).
A(n,k) = 1/n * A277537(n,k).

cp's OEIS Frontend

A179230 n-th derivative of x^(x^x) at x=1.

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