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A154342 T(n,k) an additive decomposition of the signed tangent number (triangle read by rows). 6
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified September 22 00:25 EDT 2017. Contains 292326 sequences.