A293473 - OEIS

%I #14 Oct 11 2017 10:24:20

%S 1,1,2,4,6,4,12,30,24,8,52,144,156,80,16,240,760,1020,680,240,32,1188,

%T 4440,6720,5640,2640,672,64,6804,26712,47040,45640,26880,9408,1792,

%U 128,38960,175392,338016,376320,261520,115584,31360,4608,256

%N 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.

%e Triangle starts:

%e 0: [   1]

%e 1: [   1,     2]

%e 2: [   4,     6,     4]

%e 3: [  12,    30,    24,     8]

%e 4: [  52,   144,   156,    80,    16]

%e 5: [ 240,   760,  1020,   680,   240,   32]

%e 6: [1188,  4440,  6720,  5640,  2640,  672,   64]

%e 7: [6804, 26712, 47040, 45640, 26880, 9408, 1792, 128]

%e ...

%e 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].

%p # Function dx in A293472.

%p ListTools:-Flatten([seq(dx(2, n), n=0..8)]);

%t (* Function dx in A293472. *)

%t Table[dx[2, n], {n, 0, 7}] // Flatten

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

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

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

%Y Cf. A290268.

%K sign,tabl

%O 0,3

%A _Peter Luschny_, Oct 10 2017