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T(n,k) an additive decomposition of the signed tangent number (triangle read by rows).
6

%I #12 Sep 13 2016 02:53:44

%S 1,2,-1,4,-5,1,8,-19,9,0,16,-65,55,0,-6,32,-211,285,0,-120,30,64,-665,

%T 1351,0,-1470,810,-90,128,-2059,6069,0,-14280,13020,-3150,0

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

%C 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.

%H G. C. Greubel, <a href="/A154342/b154342.txt">Table of n, a(n) for the first 50 rows</a>

%H Peter Luschny, <a href="http://www.luschny.de/math/seq/SwissKnifePolynomials.html">The Swiss-Knife polynomials.</a>

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

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

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

%e 1,

%e 2, -1,

%e 4, -5, 1,

%e 8, -19, 9, 0,

%e 16, -65, 55, 0, -6,

%e 32, -211, 285, 0, -120, 30,

%e 64, -665, 1351, 0, -1470, 810, -90,

%e 128, -2059, 6069, 0, -14280, 13020, -3150, 0

%p 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);

%t 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 *)

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

%K easy,sign,tabl

%O 0,2

%A _Peter Luschny_, Jan 07 2009