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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
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-François Alcover, Jul 30 2013, after Maple *)
CROSSREFS
Sequence in context: A065625 A287213 A284631 * A348863 A202181 A130749
KEYWORD
easy,sign,tabl
AUTHOR
Peter Luschny, Jan 07 2009
STATUS
approved

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Last modified March 29 06:34 EDT 2024. Contains 371265 sequences. (Running on oeis4.)