Original entry on oeis.org
1, 2, 5, 13, 37, 106, 331, 981, 3473, 9010, 49481, -7435, 1744909, -13392950, 186659383, -2369054219, 33839782689, -510323573086, 8221794054733, -140449867800547, 2538204766893461, -48376680944601302, 969915363800997571, -20407191628360339979
Offset: 0
Since p(3, t) = 3 + 6*t + 3*t^2 + t^3 (compare the example in A293472), a(3) = p(3, 1) = 13.
A293474
Triangle read by rows, coefficients of polynomials in t = log(x) of the n-th derivative of x^(x^3), evaluated at x = 1. T(n, k) with n >= 0 and 0 <= k <= n.
Original entry on oeis.org
1, 1, 3, 6, 12, 9, 27, 78, 81, 27, 156, 564, 720, 432, 81, 1110, 4320, 6930, 5400, 2025, 243, 8322, 37260, 68940, 66420, 34830, 8748, 729, 70098, 347382, 722610, 824040, 541485, 200718, 35721, 2187
Offset: 0
Triangle start:
0: [ 1]
1: [ 1, 3]
2: [ 6, 12, 9]
3: [ 27, 78, 81, 27]
4: [ 156, 564, 720, 432, 81]
5: [ 1110, 4320, 6930, 5400, 2025, 243]
6: [ 8322, 37260, 68940, 66420, 34830, 8748, 729]
7: [70098, 347382, 722610, 824040, 541485, 200718, 35721, 2187]
...
For n = 3, the 3rd derivative of x^(x^3) is p(3,x,t) = 27*t^3*x^6*x^(x^3) + 27*t^2*x^6*x^(x^3) + 9*t*x^6*x^(x^3) + x^6*x^(x^3) + 54*t^2*x^3*x^(x^3) + 63*t*x^3*x^(x^3) + 15*x^3*x^(x^3) + 6*t*x^(x^3) + 11*x^(x^3) where log(x) is substituted by t. Evaluated at x = 1: p(3,1,t) = 27 + 78*t + 81*t^2 + 27*t^3 with coefficients [27, 78, 81, 27].
More generally, consider the n-th derivative of x^(x^m).
-
# Function dx in A293472.
ListTools:-Flatten([seq(dx(3, n), n=0..8)]);
-
(* Function dx in A293472. *)
Table[dx[3, n], {n, 0, 7}] // Flatten
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.
Original entry on oeis.org
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
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].
More generally, consider the n-th derivative of x^(x^m).
-
# Function dx in A293472.
ListTools:-Flatten([seq(dx(2, n), n=0..8)]);
-
(* Function dx in A293472. *)
Table[dx[2, n], {n, 0, 7}] // Flatten
Showing 1-3 of 3 results.