A239322 -id:A239322 - cp's OEIS frontend

A240559 a(n) = -2^n(E(n, 1/2) + E(n, 1) + (n mod 2)2*(E(n+1, 1/2) + E(n+1, 1))), where E(n, x) are the Euler polynomials.

0, 0, 1, -3, -5, 45, 61, -1113, -1385, 42585, 50521, -2348973, -2702765, 176992725, 199360981, -17487754833, -19391512145, 2195014332465, 2404879675441, -341282303124693, -370371188237525, 64397376340013805, 69348874393137901, -14499110277050234553

Offset: 0

Peter Luschny, Apr 17 2014

sign

			G.f. = x^2 - 3*x^3 - 5*x^4 + 45*x^5 + 61*x^6 - 1113*x^7 - 1385*x^8 + ...

L. Seidel, Über eine einfache Entstehungsweise der Bernoullischen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, Vol. 7 (1877), 157-187.

Cf. A000364, A147315, A186370, A241242.

Cf. A239005, A239322.

A240559 := proc(n) euler(n,1/2) + euler(n,1); if n mod 2 = 1 then % + 2*(euler(n+1,1/2)+euler(n+1,1)) fi; -2^n*% end: seq(A240559(n),n=0..19);

skp[n_, x_] := Sum[Binomial[n, k]*EulerE[k]*x^(n-k), {k, 0, n}]; skp[n_, x0_?NumericQ] := skp[n, x] /. x -> x0; a[n_] := Sum[(-1)^(n-k)*Binomial[n, k]*(skp[k, 0] + skp[k+1, -1]), {k, 0, n}]; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Dec 09 2014, after Peter Luschny *)

# Efficient computation with L. Seidel's boustrophedon transformation.
def A240559_list(n) :
    A = [0]*(n+1); A[0] = 1; R = [0]
    k = 0; e = 1; x = -1; s = -1
    for i in (0..n):
        Am = 0; A[k + e] = 0; e = -e;
        for j in (0..i): Am += A[k]; A[k] = Am; k += e
        if e == 1: x += 1; s = -s
        v = -A[-x] if e == 1 else A[-x] - A[x]
        if i > 1: R.append(s*v)
    return R
A240559_list(24)

a(2*n) = (-1)^(n+1)*A147315(2*n,1) = (-1)^(n+1)*A186370(2*n,2*n) =(-1)^(n+1)*A000364(n) for n>0.
a(2*n+1) = (-1)^n*A147315(2*n+1,2) = (-1)^n*A186370(2*n,2*n-1) = A241242(n).
a(n) = Sum_{k=0..n} (-1)^(n-k)*2^k*binomial(n,k)*(E(k,1/2) + 2*E(k+1,0)) where E(n,x) are the Euler polynomials.
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*(skp(k,0) + skp(k+1,-1)), where skp(n, x) are the Swiss-Knife polynomials A153641.
a(n) = A239322(n) + A239005(n+1) - A239005(n). - Paul Curtz, Apr 18 2014
E.g.f.: 1 - sech(x) - tanh(x) + sinh(x)*sech(x)^2 = ((exp(-x)-1)*sech(x))^2 / 2. - Sergei N. Gladkovskii, Nov 20 2014
E.g.f.: (1 - sech(x)) * (1 - tanh(x)). - Michael Somos, Nov 22 2014

A241209 a(n) = E(n) - E(n+1), where E(n) are the Euler numbers A122045(n).

Original entry on oeis.org

1, 1, -1, -5, 5, 61, -61, -1385, 1385, 50521, -50521, -2702765, 2702765, 199360981, -199360981, -19391512145, 19391512145, 2404879675441, -2404879675441, -370371188237525, 370371188237525, 69348874393137901, -69348874393137901, -15514534163557086905

Offset: 0

Paul Curtz, Apr 17 2014

sign

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, ...).

G. C. Greubel, Table of n, a(n) for n = 0..475

Cf. A000364, A000657, A008282, A016116, A046978, A119880, A122045, A153641, A155585, A214247, A214267, A239005, A239322.

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

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

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 *)

[euler_number(n) - euler_number(n+1) for n in range(41)] # G. C. Greubel, Jun 07 2023

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

cp's OEIS Frontend

A240559 a(n) = -2^n(E(n, 1/2) + E(n, 1) + (n mod 2)2*(E(n+1, 1/2) + E(n+1, 1))), where E(n, x) are the Euler polynomials.

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A241209 a(n) = E(n) - E(n+1), where E(n) are the Euler numbers A122045(n).

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cp's OEIS Frontend

A240559 a(n) = -2^n*(E(n, 1/2) + E(n, 1) + (n mod 2)*2*(E(n+1, 1/2) + E(n+1, 1))), where E(n, x) are the Euler polynomials.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A241209 a(n) = E(n) - E(n+1), where E(n) are the Euler numbers A122045(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

SageMath

Formula

A240559 a(n) = -2^n(E(n, 1/2) + E(n, 1) + (n mod 2)2*(E(n+1, 1/2) + E(n+1, 1))), where E(n, x) are the Euler polynomials.