A392124 Irregular triangular array read by rows: row n shows the coefficients in the expansion of the n-th derivative of tan(x) as a linear combination of products of powers of sec(x)^2 and tan(x).

1, 1, 2, 2, 4, 16, 8, 16, 88, 16, 272, 416, 32, 272, 2880, 1824, 64, 7936, 24576, 7680, 128, 7936, 137216, 185856, 31616, 256, 353792, 1841152, 1304832, 128512, 512, 353792, 9061376, 21253376, 8728576, 518656, 1024

Offset

1,3

Comments

Reversing each of the rows gives essentially the array in A008303, where, in the e.g.f. item in the Formula section (and only there), reference is made to derivatives of tan(x) and sec(x).

Links

Table of n, a(n) for n=1..37.

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

First 11 rows:
     1
     1
     2
     2      4
    16      8
    16     88     16
   272    416     32
   272   2880   1824    64
  7936  24576   7680   128
  7936 137216 185856 31616 256
Row 6 represents (5th derivative of tan(x)) = 16*sec(x)^6 + 88*sec(x)^4*tan(x)^2 + 16*sec(x)^2*tan(x)^4.
Row 7 represents (6th derivative of tan(x)) = 272*sec(x)^6*tan(x) + 416*sec(x)^4*tan(x)^3 + 32*sec(x)^2*tan(x)^5.

Mathematica

Table[Expand[D[Tan[x], {x, n}]], {n, 0, 9}] // Column

Crossrefs

Cf. A000182 (column 1), A008303, A155100, A392676.

Sequence in context: A176190 A106241 A395972 * A387476 A153991 A153994

Adjacent sequences: A392121 A392122 A392123 * A392125 A392126 A392127

Keyword

nonn,tabf

Author

Clark Kimberling, Jan 19 2026

Status

approved