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A241209 a(n) = E(n) - E(n+1), where E(n) are the Euler numbers A122045(n). 2
1, 1, -1, -5, 5, 61, -61, -1385, 1385, 50521, -50521, -2702765, 2702765, 199360981, -199360981, -19391512145, 19391512145, 2404879675441, -2404879675441, -370371188237525, 370371188237525, 69348874393137901, -69348874393137901, -15514534163557086905 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

A version of the Seidel triangle (1877) for the integer Euler numbers is

              1

           1    1

         2    2   1

       2   4    5   5

    16  16   14  10   5

  16  32  46   56  61  61

etc.

It is not in the OEIS. See A008282.

The first diagonal, Es(n)=1, 1, 1, 5, 5, 61, 61, 1385, 1385, ..., comes from essentially A000364(n) repeated.

a(n) is Es(n) signed two by two.

Difference table of a(n):

1,       1,   -1,   -5,     5,    61,   -61, -1385, ...

0,      -2,   -4,   10,    56,  -122, -1324, ...

-2,     -2,   14,   46,  -178, -1202, ...

0,      16,   32, -224, -1024, ...

16,     16, -256, -800, ...

0,    -272, -544, ...

-272, -272, ...

0,

etc.

Sum of the antidiagonals: 1, 1, -5, -11, 61, 211, -385, ... = A239322(n+1).

Main diagonal interleaved with the first upper diagonal: 1, 1, -2, -4, 14, 46, ... = signed A214267(n+1).

Inverse binomial transform (first column): A155585(n+1).

The Akiyama-Tanigawa transform applied to A046978(n+1)/A016116(n) gives

1,     1,   1/2,   0, -1/4, -1/4, -1/8, 0,...

0,     1,   3/2,   1,    0, -3/4, -7/8,...

-1,   -1,   3/2,   4, 15/4,  3/4,...

0,    -5, -15/2,   1,   15,...

5,     5, -51/2, -56,...

0,    61, 183/2,...

-61, -61,...

0,...

etc.

A122045(n) and A239005(n) are reciprocal sequences by their inverse binomial transform. In their respective difference table, two different signed versions of A214247(n) appear: 1) interleaved main diagonal and first under diagonal (1, -1, -1, 2, 4, -14,...) and 2) interleaved main diagonal and first upper diagonal (1, 1, -1, -2, 4, 14,...).

LINKS

Table of n, a(n) for n=0..23.

FORMULA

a(n) = A119880(n+1) - A119880(n).

a(n) is the second column of the fractional array.

a(n) = (-1)^n*second column of the array in A239005(n).

a(n) = skp(n, 0) - skp(n+1, 0), where skp(n, x) are the Swiss-Knife polynomials A153641. - Peter Luschny, Apr 17 2014

E.g.f.: exp(x)/cosh(x)^2. - Sergei N. Gladkovskii, Jan 23 2016

G.f. T(0)/x-1/x, where T(k) = 1-x*(k+1)/(x*(k+1)-(1-x)/(1-x*(k+1)/(x*(k+1)+(1-x)/T(k+1)))). - Sergei N. Gladkovskii, Jan 23 2016

MAPLE

A241209 := proc(n) local v, k, h, m; m := `if`(n mod 2 = 0, n, n+1);

h := k -> `if`(k mod 4 = 0, 0, (-1)^iquo(k, 4));

(-1)^n*add(2^iquo(-k, 2)*h(k+1)*add((-1)^v*binomial(k, v)*(v+1)^m, v=0..k)

, k=0..m) end: seq(A241209(n), n=0..24); # Peter Luschny, Apr 17 2014

MATHEMATICA

skp[n_, x_] := Sum[Binomial[n, k]*EulerE[k]*x^(n-k), {k, 0, n}]; a[n_] := skp[n, x] - skp[n+1, x] /. x -> 0; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Apr 17 2014, after Peter Luschny *)

Table[EulerE[n] - EulerE[n + 1], {n, 0, 30}] (* Vincenzo Librandi, Jan 24 2016 *)

-Differences/@Partition[EulerE[Range[0, 30]], 2, 1]//Flatten (* Harvey P. Dale, Apr 16 2019 *)

CROSSREFS

Cf. A000657.

Sequence in context: A009390 A009334 A151467 * A171726 A129358 A318420

Adjacent sequences:  A241206 A241207 A241208 * A241210 A241211 A241212

KEYWORD

sign

AUTHOR

Paul Curtz, Apr 17 2014

STATUS

approved

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Last modified June 20 00:47 EDT 2019. Contains 324223 sequences. (Running on oeis4.)