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A154345 B(n,k) an additive decomposition of (4^n-2^n)*B(n), B(n) the Bernoulli numbers (triangle read by rows).

%I A154345 #19 Feb 03 2025 01:54:52
%S A154345 1,4,-2,12,-15,3,32,-76,36,0,80,-325,275,0,-30,192,-1266,1710,0,-720,
%T A154345 180,448,-4655,9457,0,-10290,5670,-630,1024,-16472,48552,0,-114240,
%U A154345 104160,-25200,0
%N A154345 B(n,k) an additive decomposition of (4^n-2^n)*B(n), B(n) the Bernoulli numbers (triangle read by rows).
%C A154345 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 A154345 G. C. Greubel, <a href="/A154345/b154345.txt">Table of n, a(n) for the first 50 rows</a>
%H A154345 Peter Luschny, <a href="http://www.luschny.de/math/seq/SwissKnifePolynomials.html">The Swiss-Knife polynomials.</a>
%F A154345 Let c(k) = ((-1)^floor(k/4) / 2^floor(k/2)) * [4 not div k+1] (Iverson notation).
%F A154345 B(n,k) = Sum_{v=0..k} (-1)^v*binomial(k,v)*n*c(k)*(v+2)^(n-1).
%F A154345 B(n) = (Sum_{k=0..n} B(n,k)) / (4^n-2^n).
%e A154345 Triangle begins:
%e A154345   1,
%e A154345   4,    -2,
%e A154345   12,   -15,    3,
%e A154345   32,   -76,    36,    0,
%e A154345   80,   -325,   275,   0, -30,
%e A154345   192,  -1266,  1710,  0, -720,    180,
%e A154345   448,  -4655,  9457,  0, -10290,  5670,   -630,
%e A154345   1024, -16472, 48552, 0, -114240, 104160, -25200, 0,
%e A154345   ...
%p A154345 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 A154345 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 A154345 Cf. A153641, A154341, A154342, A154343, A154344.
%K A154345 easy,sign,tabl
%O A154345 0,2
%A A154345 _Peter Luschny_, Jan 07 2009