%I #12 Sep 13 2016 02:53:19
%S 1,1,-1,1,-3,1,1,-7,6,0,1,-15,25,0,-6,1,-31,90,0,-90,30,1,-63,301,0,
%T -840,630,-90,1,-127,966,0,-6300,7980,-2520,0,1,-255,3025,0,-41706,
%U 79380,-41580,0,2520
%N E(n,k), an additive decomposition of the Euler number (triangle read by rows).
%C The Swiss-Knife polynomials A153641 can be understood as a sum of polynomials. Evaluated at x=0 these polynomials result in a decomposition of the Euler number A122045.
%H G. C. Greubel, <a href="/A154341/b154341.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 E(n,k) = Sum_{v=0,..,k} ( (-1)^(v)*binomial(k,v)*c(k)*(v+1)^n ),
%F E(n) = Sum_{k=0,..,n} E(n,k).
%e 1,
%e 1, -1,
%e 1, -3, 1,
%e 1, -7, 6, 0,
%e 1, -15, 25, 0, -6,
%e 1, -31, 90, 0, -90, 30,
%e 1, -63, 301, 0, -840, 630, -90,
%e 1, -127, 966, 0, -6300, 7980, -2520, 0,
%e 1, -255, 3025, 0, -41706, 79380, -41580, 0, 2520.
%p E := 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+1)^n,v=0..k) end: seq(print(seq(E(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])]; e[n_, k_] := Sum[(-1)^v*Binomial[k, v]*c[k]*(v+1)^n, {v, 0, k}]; Table[e[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 30 2013, after Maple *)
%Y Cf. A153641, A154342, A154343, A154344, A154345.
%K easy,sign,tabl
%O 0,5
%A _Peter Luschny_, Jan 07 2009