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A154341 E(n,k), an additive decomposition of the Euler number (triangle read by rows). 6
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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.

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).

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

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

EXAMPLE

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.

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-Fran├žois Alcover, Jul 30 2013, after Maple *)

CROSSREFS

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

Sequence in context: A065625 A287213 A284631 * A202181 A130749 A250118

Adjacent sequences:  A154338 A154339 A154340 * A154342 A154343 A154344

KEYWORD

easy,sign,tabl

AUTHOR

Peter Luschny, Jan 07 2009

STATUS

approved

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Last modified September 22 00:25 EDT 2017. Contains 292326 sequences.