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).
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
G. C. Greubel, Table of n, a(n) for n = 0..475
FORMULA
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 *)
PROG
(Magma)
EulerPoly:= func< n, x | (&+[ (&+[ (-1)^j*Binomial(k, j)*(x+j)^n : j in [0..k]])/2^k: k in [0..n]]) >;
Euler:= func< n | 2^n*EulerPoly(n, 1/2) >; // A122045
[Euler(n) - Euler(n+1): n in [0..40]]; // G. C. Greubel, Jun 07 2023
(SageMath) [euler_number(n) - euler_number(n+1) for n in range(41)] # G. C. Greubel, Jun 07 2023
CROSSREFS
KEYWORD
sign
AUTHOR
Paul Curtz, Apr 17 2014
STATUS
approved