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

%I A154342 #15 Feb 03 2025 01:54:14
%S A154342 1,2,-1,4,-5,1,8,-19,9,0,16,-65,55,0,-6,32,-211,285,0,-120,30,64,-665,
%T A154342 1351,0,-1470,810,-90,128,-2059,6069,0,-14280,13020,-3150,0
%N A154342 T(n,k) an additive decomposition of the signed tangent number (triangle read by rows).
%C A154342 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 A154342 G. C. Greubel, <a href="/A154342/b154342.txt">Table of n, a(n) for the first 50 rows</a>
%H A154342 Peter Luschny, <a href="http://www.luschny.de/math/seq/SwissKnifePolynomials.html">The Swiss-Knife polynomials.</a>
%F A154342 Let c(k) = ((-1)^floor(k/4) / 2^floor(k/2)) * [4 not div k+1] (Iverson notation).
%F A154342 T(n,k) = Sum_{v=0..k} (-1)^v*binomial(k,v)*c(k)*(v+2)^n.
%F A154342 A155585(n) = Sum_{k=0..n} T(n,k).
%e A154342 Triangle begins:
%e A154342     1,
%e A154342     2,    -1,
%e A154342     4,    -5,    1,
%e A154342     8,   -19,    9, 0,
%e A154342    16,   -65,   55, 0,     -6,
%e A154342    32,  -211,  285, 0,   -120,    30,
%e A154342    64,  -665, 1351, 0,  -1470,   810,   -90,
%e A154342   128, -2059, 6069, 0, -14280, 13020, -3150, 0,
%e A154342   ...
%p A154342 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 A154342 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 A154342 Cf. A009006, A155585.
%Y A154342 Cf. A153641, A154341, A154343, A154344, A154345.
%K A154342 easy,sign,tabl
%O A154342 0,2
%A A154342 _Peter Luschny_, Jan 07 2009