A154342 T(n,k) an additive decomposition of the signed tangent number (triangle read by rows).

1, 2, -1, 4, -5, 1, 8, -19, 9, 0, 16, -65, 55, 0, -6, 32, -211, 285, 0, -120, 30, 64, -665, 1351, 0, -1470, 810, -90, 128, -2059, 6069, 0, -14280, 13020, -3150, 0

Offset

0,2

Comments

The Swiss-Knife polynomials A153641 can be understood as a sum of polynomials. Evaluated at x=1 these polynomials result in a decomposition of the signed tangent numbers A009006.

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows

Peter Luschny, The Swiss-Knife polynomials.

Formula

Let c(k) = frac{(-1)^{floor(k/4)}{2^{floor(k/2)}} [4 not div k] (Iverson notation).

T(n,k) = Sum_{v=0,..,k} ( (-1)^(v)*binomial(k,v)*c(k)*(v+2)^n );

T(n) = Sum_{k=0,..,n} T(n,k).

Example

  1,
  2,    -1,
  4,    -5,    1,
  8,   -19,    9, 0,
 16,   -65,   55, 0,     -6,
 32,  -211,  285, 0,   -120,    30,
 64,  -665, 1351, 0,  -1470,   810,   -90,
128, -2059, 6069, 0, -14280, 13020, -3150, 0

Maple

T := proc(n, k) local v, c; c := m -> if irem(m+1, 4) = 0 then 0 else 1/((-1)^iquo(m+1, 4)*2^iquo(m, 2)) fi; add((-1)^(v)*binomial(k, v)*c(k)*(v+2)^n, v=0..k) end: seq(print(seq(T(n, k), k=0..n)), n=0..8);

Mathematica

c[m_] := If[Mod[m+1, 4] == 0, 0, 1/((-1)^Quotient[m+1, 4]*2^Quotient[m, 2])]; t[n_, k_] := Sum[(-1)^v*Binomial[k, v]*c[k]*(v+2)^n, {v, 0, k}]; Table[t[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 30 2013, after Maple *)

Crossrefs

Cf. A153641, A154341, A154343, A154344, A154345.

Sequence in context: A114164 A176667 A126182 * A143494 A124960 A137346

Adjacent sequences: A154339 A154340 A154341 * A154343 A154344 A154345

Keyword

easy,sign,tabl

Author

Peter Luschny, Jan 07 2009

Status

approved