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A154343 S(n,k) an additive decomposition of the Springer number (generalized Euler number), (triangle read by rows). 6
1, 3, -2, 9, -16, 4, 27, -98, 60, 0, 81, -544, 616, 0, -96, 243, -2882, 5400, 0, -3360, 960, 729, -14896, 43564, 0, -72480, 46080, -5760, 2187, -75938, 334740, 0, -1246560, 1323840, -362880, 0, 6561, -384064, 2495056, 0, -18801216, 29675520 (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=1/2 and multiplied by 2^n these polynomials result in a decomposition of the Springer numbers A001586.
LINKS
FORMULA
Let c(k) = frac{(-1)^{floor(k/4)}{2^{floor(k/2)}} [4 not div k] (Iverson notation).
S(n,k) = Sum_{v=0,..,k} ( (-1)^(v)*binomial(k,v)*2^n*c(k)*(v+3/2)^n );
S(n) = Sum_{k=0,..,n} S(n,k).
EXAMPLE
1,
3, -2,
9, -16, 4,
27, -98, 60, 0,
81, -544, 616, 0, -96,
243, -2882, 5400, 0, -3360, 960,
729, -14896, 43564, 0, -72480, 46080, -5760,
2187, -75938, 334740, 0, -1246560, 1323840, -362880, 0,
6561, -384064, 2495056, 0, -18801216, 29675520, -13386240, 0, 645120.
MAPLE
S := 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)*2^n*c(k)*(v+3/2)^n, v=0..k) end: seq(print(seq(S(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])]; s[n_, k_] := Sum[(-1)^v*Binomial[k, v]*2^n*c[k]*(v+3/2)^n, {v, 0, k}]; Table[s[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 30 2013, after Maple *)
CROSSREFS
Sequence in context: A358603 A076584 A309673 * A049969 A088634 A118791
KEYWORD
easy,sign,tabl
AUTHOR
Peter Luschny, Jan 07 2009
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)