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%I #15 Sep 13 2016 02:54:17
%S 1,4,-2,12,-15,3,32,-76,36,0,80,-325,275,0,-30,192,-1266,1710,0,-720,
%T 180,448,-4655,9457,0,-10290,5670,-630,1024,-16472,48552,0,-114240,
%U 104160,-25200,0
%N B(n,k) an additive decomposition of (4^n-2^n)*B(n), B(n) the Bernoulli numbers (triangle read by rows).
%C The Swiss-Knife polynomials A153641 can be understood as a sum of polynomials. Evaluated at x=1 and multiplied by n this results in a decomposition of (4^n-2^n) times the Bernoulli numbers A027641/A027642 (for n>0 and B_1 = 1/2).
%H G. C. Greubel, <a href="/A154345/b154345.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 B(n,k) = Sum(v=0..k,(-1)^(v)*binomial(k,v)*n*c(k)*(v+2)^(n-1));
%F B(n) = (Sum(k=0..n, B(n,k)) / (4^n-2^n)
%e 1,
%e 4, -2,
%e 12, -15, 3,
%e 32, -76, 36, 0,
%e 80, -325, 275, 0, -30,
%e 192, -1266, 1710, 0, -720, 180,
%e 448, -4655, 9457, 0, -10290, 5670, -630,
%e 1024, -16472, 48552, 0, -114240, 104160, -25200, 0.
%p B := 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)*n*c(k)*(v+2)^(n-1),v=0..k) end: seq(print(seq(B(n,k),k=0..(n-1))),n=0..8);
%t c[m_] := If[Mod[m+1, 4] == 0, 0, 1/((-1)^Quotient[m+1, 4]*2^Quotient[m, 2])]; b[n_, k_] := Sum[(-1)^v*Binomial[k, v]*n*c[k]*(v+2)^(n-1), {v, 0, k}]; Table[b[n, k], {n, 0, 8}, {k, 0, n - 1}] // Flatten (* _Jean-François Alcover_, Jul 30 2013, after Maple *)
%Y Cf. A153641, A154341, A154342, A154343, A154344.
%K easy,sign,tabl
%O 0,2
%A _Peter Luschny_, Jan 07 2009
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