cp's OEIS Frontend

A154341 E(n,k), an additive decomposition of the Euler number (triangle read by rows).

%I A154341 #18 Feb 03 2025 00:13:33
%S A154341 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 A154341 -840,630,-90,1,-127,966,0,-6300,7980,-2520,0,1,-255,3025,0,-41706,
%U A154341 79380,-41580,0,2520
%N A154341 E(n,k), an additive decomposition of the Euler number (triangle read by rows).
%C A154341 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 A154341 G. C. Greubel, <a href="/A154341/b154341.txt">Table of n, a(n) for the first 50 rows</a>
%H A154341 Peter Luschny, <a href="http://www.luschny.de/math/seq/SwissKnifePolynomials.html">The Swiss-Knife polynomials.</a>
%F A154341 Let c(k) = ((-1)^floor(k/4) / 2^floor(k/2)) * [4 not div k+1] (Iverson notation).
%F A154341 E(n,k) = Sum_{v=0..k} (-1)^v*binomial(k,v)*c(k)*(v+1)^n,
%F A154341 A122045(n) = Sum_{k=0..n} E(n,k).
%e A154341 Triangle begins:
%e A154341   1,
%e A154341   1,   -1,
%e A154341   1,   -3,    1,
%e A154341   1,   -7,    6, 0,
%e A154341   1,  -15,   25, 0,     -6,
%e A154341   1,  -31,   90, 0,    -90,    30,
%e A154341   1,  -63,  301, 0,   -840,   630,    -90,
%e A154341   1, -127,  966, 0,  -6300,  7980,  -2520,  0,
%e A154341   1, -255, 3025, 0, -41706, 79380, -41580,  0, 2520,
%e A154341   ...
%p A154341 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 A154341 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 A154341 Cf. A122045, A153641, A154342, A154343, A154344, A154345.
%K A154341 easy,sign,tabl
%O A154341 0,5
%A A154341 _Peter Luschny_, Jan 07 2009