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

Author

Peter Luschny, Jan 07 2009

Keywords

• easy
• sign
• tabl

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.

Examples

			Triangle begins:
    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,
  ...
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows
• Peter Luschny, The Swiss-Knife polynomials.

Crossrefs

Cf. A009006, A155585.

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

Programs

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 *)

Formula

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

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

A155585(n) = Sum_{k=0..n} T(n,k).