A392676 Irregular triangular array read by rows: row n shows the coefficients of the irreducible polynomial p(u) when the expansion of the n-th derivative of tan(x) is expressed in the form 2^q*(1 + u^2)*p(u) if n is odd, and 2^q*u*(1 + u^2)*p(u) if n is even, where u = tan(x).

1, 1, 1, 3, 2, 3, 2, 15, 15, 17, 60, 45, 17, 231, 525, 315, 62, 378, 630, 315, 62, 1320, 5040, 6615, 2835, 1382, 12720, 34965, 37800, 14175, 1382, 42306, 238425, 509355, 467775, 155925, 21844, 280731, 1121670, 1954260, 1559250, 467775, 21844, 907725, 7012005

Offset

1,4

Links

Table of n, a(n) for n=1..45.

Michael E. Hoffman, Derivative Polynomials for Tangent and Secant, American Mathematical Monthly, 102 (1995), 23-30.

Feng Qi, Derivatives of tangent function and tangent numbers, Applied Mathematics and Computation, 268 (2015), 844-858.

Example

Triangle begins:
     1
     1
     1      3
     2      3
     2     15      15
    17     60      45
    17    231     525     315
    62    378     630     315
    62   1320    5040    6615    2835
  1382  12720   34965   37800   14175
  1382  42306  238425  509355  467775  155925
Row 6 represents (5th derivative of tan(x)) = 16*(1 + u^2)*(17 + 231*u^2 + 525*u^4 + 315*u^6).
Row 7 represents (6th derivative of tan(x)) = 128*u*(1 + u^2)*(62 + 378*u^2 + 630*u^4 + 315*u^6).

Mathematica

Map[Factor[Expand[D[Tan[x], {x, #}] /. Sec[x]^y_?EvenQ -> (1 + Tan[x]^2)^(y/2) /. Tan[x] -> u]] &, Range[15]]
(* Peter J. C. Moses, Nov 08 2025 *)

Crossrefs

Cf. A002430 (column 1), A392124.

Sequence in context: A362688 A214254 A370903 * A153092 A165601 A324030

Adjacent sequences: A392673 A392674 A392675 * A392677 A392678 A392679

Keyword

nonn,tabf

Author

Clark Kimberling, Jan 19 2026

Extensions

More terms from Sean A. Irvine, Feb 08 2026

Status

approved