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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). 6
 1, 4, -2, 12, -15, 3, 32, -76, 36, 0, 80, -325, 275, 0, -30, 192, -1266, 1710, 0, -720, 180, 448, -4655, 9457, 0, -10290, 5670, -630, 1024, -16472, 48552, 0, -114240, 104160, -25200, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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). LINKS G. C. Greubel, Table of n, a(n) for the first 50 rows Peter Luschny, The Swiss-Knife polynomials. FORMULA Let c(k) = frac{(-1)^{floor(k/4)}{2^{floor(k/2)}} [4 not div k] (Iverson notation). B(n,k) = Sum(v=0..k,(-1)^(v)*binomial(k,v)*n*c(k)*(v+2)^(n-1)); B(n) = (Sum(k=0..n, B(n,k)) / (4^n-2^n) EXAMPLE 1, 4,    -2, 12,   -15,    3, 32,   -76,    36,    0, 80,   -325,   275,   0, -30, 192,  -1266,  1710,  0, -720,    180, 448,  -4655,  9457,  0, -10290,  5670,   -630, 1024, -16472, 48552, 0, -114240, 104160, -25200,  0. MAPLE 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); MATHEMATICA 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 *) CROSSREFS Cf. A153641, A154341, A154342, A154343, A154344. Sequence in context: A111667 A019239 A143944 * A058095 A105196 A167557 Adjacent sequences:  A154342 A154343 A154344 * A154346 A154347 A154348 KEYWORD easy,sign,tabl AUTHOR Peter Luschny, Jan 07 2009 STATUS approved

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Last modified September 21 21:21 EDT 2018. Contains 315262 sequences. (Running on oeis4.)