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

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

Offset: 0

Author

Peter Luschny, Jan 07 2009

Keywords

• easy
• sign
• tabl

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).

Examples

			Triangle begins:
  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,
  ...
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows
• Peter Luschny, The Swiss-Knife polynomials.

Crossrefs

Cf. A153641, A154341, A154342, A154343, A154344.

Programs

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 *)

Formula

Let c(k) = ((-1)^floor(k/4) / 2^floor(k/2)) * [4 not div k+1] (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).