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

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, -840, 630, -90, 1, -127, 966, 0, -6300, 7980, -2520, 0, 1, -255, 3025, 0, -41706, 79380, -41580, 0, 2520

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=0 these polynomials result in a decomposition of the Euler number A122045.

Examples

			Triangle begins:
  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,   -840,   630,    -90,
  1, -127,  966, 0,  -6300,  7980,  -2520,  0,
  1, -255, 3025, 0, -41706, 79380, -41580,  0, 2520,
  ...
		

Links

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

Crossrefs

Cf. A122045, A153641, A154342, A154343, A154344, A154345.

Programs

Maple

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);

Mathematica

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 *)

Formula

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

E(n,k) = Sum_{v=0..k} (-1)^v*binomial(k,v)*c(k)*(v+1)^n,

A122045(n) = Sum_{k=0..n} E(n,k).