A293473 Triangle read by rows, coefficients of polynomials in t = log(x) of the n-th derivative of x^(x^2), evaluated at x = 1. T(n, k) with n >= 0 and 0 <= k <= n.

1, 1, 2, 4, 6, 4, 12, 30, 24, 8, 52, 144, 156, 80, 16, 240, 760, 1020, 680, 240, 32, 1188, 4440, 6720, 5640, 2640, 672, 64, 6804, 26712, 47040, 45640, 26880, 9408, 1792, 128, 38960, 175392, 338016, 376320, 261520, 115584, 31360, 4608, 256

Offset: 0

Author

Peter Luschny, Oct 10 2017

Keywords

• sign
• tabl

Examples

			Triangle starts:
0: [   1]
1: [   1,     2]
2: [   4,     6,     4]
3: [  12,    30,    24,     8]
4: [  52,   144,   156,    80,    16]
5: [ 240,   760,  1020,   680,   240,   32]
6: [1188,  4440,  6720,  5640,  2640,  672,   64]
7: [6804, 26712, 47040, 45640, 26880, 9408, 1792, 128]
...
For n = 3, the 3rd derivative of x^(x^2) is p(3,x,t) = 8*t^3*x^3*x^(x^2) + 12*t^2*x^3*x^(x^2) + 6*t*x^3*x^(x^2) + 12*t^2*x*x^(x^2) + x^3*x^(x^2) + 24*t*x*x^(x^2) + 9*x*x^(x^2) + 2*x^(x^2)/x where log(x) is substituted by t. Evaluated at x = 1: p(3,1,t) = 12 + 30*t + 24*t^2 + 8*t^3 with coefficients [12, 30, 24, 8].
		

Crossrefs

T(n, 0) = A215524, T(n, n) = A000079.

More generally, consider the n-th derivative of x^(x^m).

A293472 (m=1), this seq. (m=2), A293474 (m=3).

Cf. A290268.

Programs

Maple

# Function dx in A293472.
ListTools:-Flatten([seq(dx(2, n), n=0..8)]);

Mathematica

(* Function dx in A293472. *)
Table[dx[2, n], {n, 0, 7}] // Flatten