A198631 -id:A198631 - cp's OEIS frontend

A342320 Integers k such that Euler(k, 1) is an integer multiple of Bernoulli(k + 1, 1).

0, 1, 5, 17, 41, 53, 125, 161, 293, 341, 377, 485, 881, 1025, 1133, 1313, 1457, 1805, 2057, 2393, 2645, 3077, 3401, 3941, 4373, 5333, 5417, 6173, 6497, 7181, 7937, 9197, 9233, 10205, 11825, 12641, 13121, 14153, 14405, 16001, 16253, 16757, 18521, 19493, 21545

Offset: 0

Peter Luschny, Mar 24 2021

nonn

			Let E(n) = Euler(n, 1) and B(n) = Bernoulli(n, 1).
2*E(0)  = 4*B(1) = 2;
2*E(1)  = 6*B(2) = 1;
2*E(5)  = 42*B(6) = 1;
2*E(17) = 58254*B(18) = 3202291;
2*E(41) = 418861572486*B(42) = 352552873457246307069012458671.

a(n) = A015942(n-1)-1 for n>=2, (a(n)+1)/2 = A014945(n) for n>=1.
a(n) = A014741(n+1) - 1. - Vaclav Kotesovec, Mar 24 2021
Cf. A341759 (subsequence of primes), A198631/A006519 (Euler), A164555/A027642 (Bernoulli).

Join[{0}, Select[Range[1000], BernoulliB[#+1, 1] != 0 && IntegerQ[EulerE[#, 1]/BernoulliB[#+1, 1]] &]] (* Vaclav Kotesovec, Mar 24 2021 *)
Select[Range[100000], IntegerQ[(2*(-1 + 2^#))/#] & ] - 1 (* Vaclav Kotesovec, Mar 24 2021 *)
L342320 := Select[Range[0, 10000], Divisible[2^(# + 2) - 2, # + 1] &];
A342320[n_] := L342320[[n + 1]] (* Peter Luschny, Apr 10 2021 *)

Numbers k such that k + 1 divides 2^(k + 2) - 2. - Vaclav Kotesovec, Mar 24 2021

A227577 Square array read by antidiagonals, A(n,k) the numerators of the elements of the difference table of the Euler polynomials evaluated at x=1, for n>=0, k>=0.

Original entry on oeis.org

1, -1, 1, 0, -1, 0, 1, 1, -1, -1, 0, 1, 1, 1, 0, -1, -1, -1, 1, 1, 1, 0, -1, -1, -5, -1, -1, 0, 17, 17, 13, 5, -5, -13, -17, -17, 0, 17, 17, 47, 13, 47, 17, 17, 0, -31, -31, -107, -73, -13, 13, 73, 107, 31, 31, 0, -31, -31, -355

Offset: 0

Paul Curtz, Jul 16 2013

sign

The difference table of the Euler polynomials evaluated at x=1:
    1,   1/2,     0,    -1/4,     0,     1/2,      0,      -17/8, ...
  -1/2, -1/2,   -1/4,    1/4,    1/2,   -1/2,   -17/8,      17/8, ...
    0,   1/4,    1/2,    1/4;    -1,   -13/8,    17/4,     107/8, ...
   1/4,  1/4,   -1/4,   -5/4,   -5/8,   47/8,    73/8,    -355/8, ...
    0,  -1/2,    -1,     5/8    13/2,   13/4,  -107/2,    -655/8, ...
  -1/2, -1/2,   13/8,   47/8,  -13/4, -227/4,  -227/8,    5687/8, ...
    0,  17/8,   17/4,  -73/8, -107/2,  227/8,  2957/4,    2957/8, ...
  17/8, 17/8, -107/8, -355/8,  655/8, 5687/8, -2957/8, -107125/8, ...
To compute the difference table, take
    1,   1/2;
  -1/2;
The next term is always half of the sum of the antidiagonals. Hence (-1/2 + 1/2 = 0)
    1,   1/2,   0;
  -1/2, -1/2;
    0;
The first column (inverse binomial transform) lists the numbers (1, -1/2, 0, 1/4, ..., not in the OEIS; corresponds to A027641/A027642). See A209308 and A060096.
A198631(n)/A006519(n+1) is an autosequence. See A181722.
Note the main diagonal: 1, -1/2, 1/2, -5/4, 13/2, -227/4, 2957/4, -107125/8, .... (See A212196/A181131.)
This twice the first upper diagonal. The autosequence is of the second kind.
From 0, -1, the algorithm gives A226158(n), full Genocchi numbers, autosequence of the first kind.
The difference table of the Bernoulli polynomials evaluated at x=1 is (apart from signs) A085737/A085738 and its analysis by Ludwig Seidel was discussed in the Luschny link. - Peter Luschny, Jul 18 2013

			Read by antidiagonals:
    1;
  -1/2,  1/2;
    0,  -1/2,   0;
   1/4,  1/4, -1/4, -1/4;
    0,   1/4,  1/2,  1/4,   0;
  -1/2, -1/2, -1/4,  1/4,  1/2,  1/2;
    0,  -1/2, - 1,  -5/4,  -1,  -1/2,   0;
  ...
Row sums: 1, 0, -1/2, 0, 1, 0, -17/4, 0, ... = 2*A198631(n+1)/A006519(n+2).
Denominators: 1, 1, 2, 1, 1, 1, 4, 1, ... = A160467(n+2)?

Peter Luschny, The computation and asymptotics of the Bernoulli numbers.
OEIS Wiki, Autosequence

Cf. A164555/A027642 in A190339.

DifferenceTableEulerPolynomials := proc(n) local A,m,k,x;
A := array(0..n,0..n); x := 1;
for m from 0 to n do for k from 0 to n do A[m,k]:= 0 od od;
for m from 0 to n do A[m,0] := euler(m,x);
   for k from m-1 by -1 to 0 do
      A[k,m-k] := A[k+1,m-k-1] - A[k,m-k-1] od od;
LinearAlgebra[Transpose](convert(A, Matrix)) end:
DifferenceTableEulerPolynomials(7);  # Peter Luschny, Jul 18 2013

t[0, 0] = 1; t[0, k_] := EulerE[k, 1]; t[n_, 0] := -t[0, n]; t[n_, k_] := t[n, k] = t[n-1, k+1] - t[n-1, k]; Table[t[n-k, k] // Numerator, {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 18 2013 *)

def DifferenceTableEulerPolynomialsEvaluatedAt1(n) :
    @CachedFunction
    def ep1(n):          # Euler polynomial at x=1
        if n < 2: return 1 - n/2
        s = add(binomial(n,k)*ep1(k) for k in (0..n-1))
        return 1 - s/2
    T = matrix(QQ, n)
    for m in range(n) :  # Compute difference table
        T[m,0] = ep1(m)
        for k in range(m-1,-1,-1) :
            T[k,m-k] = T[k+1,m-k-1] - T[k,m-k-1]
    return T
def A227577_list(m):
    D = DifferenceTableEulerPolynomialsEvaluatedAt1(m)
    return [D[k,n-k].numerator() for n in range(m) for k in (0..n)]
A227577_list(12)  # Peter Luschny, Jul 18 2013

Corrected by Jean-François Alcover, Jul 17 2013

A271573 Numerator of (0 followed by A005126(n)= 2, 4, 7, ...)/2^n.

Original entry on oeis.org

0, 1, 1, 7, 3, 21, 19, 71, 17, 265, 261, 1035, 515, 4109, 4103, 16399, 2049, 65553, 65545, 262163, 131077, 1048597, 1048587, 4194327, 1048579, 16777241, 16777229, 67108891, 33554439, 268435485, 268435471, 1073741855, 67108865, 4294967329, 4294967313

Offset: 0

Paul Curtz, Apr 10 2016

Reduced fractions: f(n) = 0, 1, 1, 7/8, 3/4, 21/32, 19/32, 71/128, 17/32, 265/512, 261/512, ... .

f(n) is an autosequence of the first kind.

			a(0), a(1), a(2), a(3), a(4), are the numerators of reduced fractions 0/1, 2/2, 4/4, 7/8, 12/16, ... .

Robert Price, Table of n, a(n) for n = 0..101

Cf. A000004, A000079, A005126, A006519, A060576(n+1), A075101, A198631, A279635 (denominator).

[0] cat [Numerator((2^(n-1)+n)/2^n): n in [1..40]]; // Vincenzo Librandi, Oct 13 2017

Prepend[Table[Numerator[(2^n + n + 1)/2^(n + 1)], {n, 0, 100}], 0] (* Robert Price, Apr 10 2016 *)
(* Computation from Oresme numbers n/2^n: *) a[n_] := Numerator[n/2^n + If[n < 2, 0, 1]/2]; (* Jean-François Alcover, Apr 28 2016, after Paul Curtz *)

a(n) = if(n==0, 0, numerator((2^(n-1)+n)/2^n)); \\ Altug Alkan, Apr 10 2016

a(n) = numerator(n/2^n + (if n<2 0 else 1)/2), a formula using Oresme numbers n/2^n. - Jean-François Alcover, Apr 28 2016 after Paul Curtz

A339058 a(n) = 4^nEuler(n, 1/4)2^(valuation_{2}(n + 1)).

Original entry on oeis.org

1, -2, -3, 44, 57, -722, -2763, 196888, 250737, -5746082, -36581523, 2049374444, 7828053417, -259141449842, -2309644635483, 705775346640176, 898621108880097, -38901437271432002, -445777636063460643, 43136210244502819244, 274613643571568682777, -14685255919931552812562

Offset: 0

Peter Luschny, Nov 27 2020

sign

			The array of the general case starts:
[k]
[1] 1,  1,  0, -1,   0,     1,     0,     -17,       0, ... [A198631]
[2] 1,  0, -1,  0,   5,     0,   -61,       0,    1385, ... [A122045]
[3] 1, -1, -2, 13,  22,  -121,  -602,   18581,   30742, ... [A156179]
[4] 1, -2, -3, 44,  57,  -722, -2763,  196888,  250737, ... [this sequence]
[5] 1, -3, -4, 99, 116, -2523, -8764, 1074243, 1242356, ... [A156182]
...

Michael De Vlieger, Table of n, a(n) for n = 0..432

Cf. A198631, A122045, A156179, A156182, A156191, A339057.

Note the difference from A001586, A188458, and A212435.

a := n -> 4^n*euler(n, 1/4)*2^padic[ordp](n+1, 2): seq(a(n), n=0..9);

Array[4^#*EulerE[#, 1/4]*2^IntegerExponent[# + 1, 2] &, 22, 0] (* Michael De Vlieger, Mar 15 2022 *)

def euler_sum(n):
    return (-1)^n*sum(2^k*binomial(n, k)*euler_number(k) for k in (0..n))
def a(n): return euler_sum(n) << valuation(n + 1, 2)
print([a(n) for n in range(22)])

A242971 Alternate n+1, 2^n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 4, 8, 5, 16, 6, 32, 7, 64, 8, 128, 9, 256, 10, 512, 11, 1024, 12, 2048, 13, 4096, 14, 8192, 15, 16384, 16, 32768, 17, 65536, 18, 131072, 19, 262144, 20, 524288, 21, 1048576, 22, 2097152, 23, 4194304, 24, 8388608, 25, 16777216, 26, 33554432

Offset: 0

Paul Curtz, Jun 22 2014

nonn

The offset 0 is a choice. Another sequence could begin with A001477 instead of A000027. The Akiyama-Tanigawa transform applied to 1/(n+1) and 1/2^n are the second Bernoulli numbers A164555(n)/A027642(n) and the second (fractional) Euler numbers A198631(n)/A006519(n+1). (The first Euler numbers are not in the OEIS). Hence a(n).
a(2n+1) - a(2n) = 2^n -n -1 = 0, 0, 1, 4, 11,... = A000295(n) (Eulerian numbers).
a(2n+1) + a(2n) = 2^n +n +1 = A005126(n).

			G.f. = 1 + x + 2*x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 4*x^6 + 8*x^7 + 5*x^8 + ...

Index entries for linear recurrences with constant coefficients, signature (0,4,0,-5,0,2).

Cf. A000027, A000079, A209308, A227577.

A242971:=n->((n+1) mod 2)*(n/2 + 1) + (n mod 2) * 2^((n-1)/2); seq(A242971(n), n=0..50); # Wesley Ivan Hurt, Jun 29 2014

Table[Mod[n + 1, 2] (n/2 + 1) + Mod[n, 2] 2^((n - 1)/2), {n, 0, 50}] (* Wesley Ivan Hurt, Jun 29 2014 *)

a(n) = ((n+1) mod 2) * (n/2 + 1) + (n mod 2) * 2^((n-1)/2). - Wesley Ivan Hurt, Jun 29 2014

G.f.: (1 + x - x^2) * (1 - x^2 - x^3) / ((1 - x^2)^2 * (1 - 2*x^2)). - Michael Somos, Jun 30 2014

A244213 Inverse binomial transform of -2 followed by A000032(n+1).

Original entry on oeis.org

-2, 3, -1, 0, 3, -7, 14, -25, 43, -72, 119, -195, 318, -517, 839, -1360, 2203, -3567, 5774, -9345, 15123, -24472, 39599, -64075, 103678, -167757, 271439, -439200, 710643, -1149847, 1860494, -3010345, 4870843, -7881192

Offset: 0

Paul Curtz, Jun 23 2014

A simple transform of a(n) is a(n) with -a(0) instead of nonzero a(0) (or -a(0) followed by a(n+1)). Example: -1 followed by A198631(n+1)/A006519(n+2). Its inverse binomial transform is -1, 3/2, -2, 9/4, -2, 3/2, -2,... = -(-1)^n*A143074(n).
Difference table of -2 followed by A000032(n+1):
-2,  1,  3,  4,  7, 11, 18,...
3,   2,  1,  3,  4,  7, 11,...
-1, -1,  2,  1,  3,  4,  7,...
0,   3, -1,  2,  1,  3,  4,...
3,  -4,  3, -1,  2,  1,  3,...
-7,  7, -4,  3, -1,  2,  1,...
14, -11, 7, -4,  3, -1,  2,...
etc.
a(n) is the first column.

Index entries for linear recurrences with constant coefficients, signature (-2,0,1).

Cf. A000045, A000032, A000204.

Vec(-(5*x^2-x-2)/((x+1)*(x^2-x-1)) + O(x^100)) \\ Colin Barker, Jun 23 2014

a(n) = -2, 3, -1, followed by -(-1)^n*A206417(n).
a(n) = (-1)^n* (A000032(n) - 4).
a(n+3) = -a(n) -(-1)^n*A022112(n).
a(n) = -2*a(n-1) + a(n-3). - Colin Barker, Jun 23 2014
G.f.: -(5*x^2-x-2) / ((x+1)*(x^2-x-1)). - Colin Barker, Jun 23 2014

A281500 Reduced denominators of f(n) = (n+1)/(2^(2+n)-2) with A026741(n+1) as numerators.

Original entry on oeis.org

2, 3, 14, 15, 62, 63, 254, 255, 1022, 1023, 4094, 4095, 16382, 16383, 65534, 65535, 262142, 262143, 1048574, 1048575, 4194302, 4194303, 16777214, 16777215, 67108862, 67108863, 268435454, 268435455, 1073741822, 1073741823, 4294967294, 4294967295, 17179869182, 17179869183

Offset: 0

Paul Curtz, Jan 23 2017

f(n) = (n+1)/A000918(n+2) = 1/2, 2/6, 3/14, 4/30, 5/62, 6/126, 7/254, 8/510, 9/1022, 10/2046, 11/4094, 12/8190, ... .
Partial reduction: 1/2, 1/3, 3/14, 2/15, 5/62, 3/63, 7/254, 4/255, 9/1022, 5/1023, 11/4094, 6/4095, ... = A026741(n+1)/a(n).
Full reduction: 1/2, 1/3, 3/14, 2/15, 5/62, 1/21, 7/254, ... = A111701(n+1)/(2, 3, 14, 15, 62, 21, ... )
A164555(n+1)/A027642(n) = 1/2, 1/6, 0, -1/30, 0, 1/42, ... = f(n) * A198631(n)/A006519(n+1) = 1, 1/2, 0, -1/4, 0, 1/2, ... .).
Via f(n), we go from the second fractional Euler numbers to the second Bernoulli numbers.
a(n) mod 10: periodic sequence of length 4: repeat [2, 3, 4, 5].
a(n) differences table:
.  2,   3, 14,  15,  62,   63, 254,  255, ...
.  1,  11,  1,  47,   1,  191,   1,  767, ... see A198693
. 10, -10, 46, -46, 190, -190, 766, -766, ... see A096045, from Bernoulli(2n).
Extension of a(n): a(-2) = -1, a(-1) = 0.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-4).

Cf. A000027, A000918, A001477, A006519, A026741, A027642, A096045, A111701, A117856, A164555, A198631, A198693, A209308, A267921.

a[n_] := (3+(-1)^n)*(2^(n+1)-1)/2; (* or *) a[n_] := If[EvenQ[n], 4^(n/2+1)-2, 4^((n+1)/2)-1]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jan 24 2017 *)

Vec((2 + 3*x + 4*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)) + O(x^50)) \\ Colin Barker, Jan 24 2017

From Colin Barker, Jan 24 2017: (Start)
G.f.: (2 + 3*x + 4*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).
a(n) = 5*a(n-2) - 4*a(n-4) for n>3. (End)
From Jean-François Alcover, Jan 24 2017: (Start)
a(n) = (3 + (-1)^n)*(2^(n + 1) - 1)/2.
a(n) = 4^((n + 1 + ((n + 1) mod 2))/2) - 1 - ((n + 1) mod 2). (End)
a(n) = a(n-2) + A117856(n+1) for n>1.
a(2*k) = 4^(k + 1) - 2, a(2*k+1) = a(2*k) + 1 = 4^(k+1) - 1.
a(2*k) + a(2*k+1) = A267921(k+1).

A232800 Denominators written by antidiagonals of interleaved A063524(n) and A002427(n)/A006955(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 2, 1, 2, 1, 6, 6, 3, 3, 6, 6, 1, 6, 3, 1, 3, 6, 1, 6, 6, 1, 3, 3, 1, 6, 6, 1, 6, 3, 3, 1, 3, 3, 6, 1, 10, 10, 15, 5, 15, 15, 5, 15, 10, 10, 1, 10, 5, 15, 15, 1, 15, 15, 5, 10, 1

Offset: 0

Paul Curtz, Nov 30 2013

The numerators of Br(n)=0, 1, 1, 1/2, 0, -1/6, 0, 1/6, 0, -3/10, 0, 5/6, 0,... are in A229979(n).
(Corresponding complementary Euler numbers Er(n):
A198631(2n+1)/A006519(2n+2)=1/2, -1/4, 1/2, -17/8, 31/2, -691/4,... =Ef2(n). (2*n+2)*Ef2(n)=1, -1, 3, -17, 155, 2073,...=-A001469(n+1),Genocchi numbers. Er(n)=interleaved (A063524(n) and -A001469(n+1)) =0, 1, 1, -1, 0, 3, 0, -17, 0, 155,... =-A226158(n).)

			1,
1, 1,
1, 1, 1,
2, 2, 2, 2,
1, 2, 1, 2, 1,
6, 6, 3, 3, 6, 6, etc.
Triangle of denominators of Br(n),complementary Bernoulli numbers,written by antidiagonals (see A229979).

A238235 Numerators of Euler twin numbers Et(n).

Original entry on oeis.org

1, -1, -1, -1, 1, 1, -1, -17, 17, 31, -31, -691, 691, 5461, -5461, -929569, 929569, 3202291, -3202291, -221930581, 221930581, 4722116521, -4722116521, -968383680827, 968383680827, 14717667114151, -14717667114151

Offset: 0

Paul Curtz, Feb 20 2014

sign

Et(n) = 1, -1/2, -1/2, -1/4, 1/4, 1/2, -1/2, -17/8, 17/8, 31/2, -31/2, -691/4, 691/4, 5461/2, -5461/2,... =a(n)/b(n) is mentioned in A233808.
Denominators: b(n)= 1, 2, 2, 4, 4, 2, 2, 8, 8,... = A065176(n) with 1 instead of 0.
Et(n) is the first difference of 0, followed by A198631(n)/A006519(n+1).
Et(n+2) = -1/2, -1/4, 1/4, 1/2,... is an autosequence of the second kind. Its main diagonal is the double of the following diagonal, the inverse binomial transform of Et(n+2) being Et(n+2) signed.
The denominators of the difference table of Et(n+2) are even numbers of the form 2^p. For the Bernoulli twin numbers A051716(n+1)/A051717(n+2), the denominators of the difference table, A168426(n), are multiples of 3.

Cf. A051716/A051717 (Bernoulli twin numbers).

Join[{1, -1, -1}, Table[{nu = Numerator[EulerE[2*n+1, 1]], -nu}, {n, 1, 12}]] // Flatten (* Jean-François Alcover, Feb 24 2014 *)

Binomial transform of A141424(n)/(A053644(n) with 1 instead of 0).

a(2n+3) = (-1)^n*A002425(n+2) = -a(2n+4).

A242246 Numerators of n*A164555(n-1)/A027642(n-1).

Original entry on oeis.org

0, 1, 1, 1, 0, -1, 0, 1, 0, -3, 0, 5, 0, -691, 0, 35, 0, -3617, 0, 43867, 0, -1222277, 0, 854513, 0, -1181820455, 0, 76977927, 0, -23749461029, 0, 8615641276005, 0, -84802531453387, 0, 90219075042845, 0

Offset: 0

Paul Curtz, May 09 2014

sign

First multiplied shifted (second) Bernoulli numbers.
A164555(n-1)/A027642(n-1) = 0 followed by (A164555(n)/A027642(n)=1, 1/2, 1/6,...) = f(n) = 0, 1, 1/2, 1/6, 0,... .
f(n+1) - f(n) = A051716(n)/A051717(n).
Generally we consider a transform applied to the autosequences of first or second kind. An autosequence is a sequence which has its inverse binomial transform equal to the signed sequence. It is of the first kind if the main diagonal is A000004=0's. It is of the second kind if the main diagonal is the first upper diagonal multiplied by 2. A000045(n) is an autosequence of the first kind. A164555(n)/A027642(n) is an autosequence of the second kind. See A190339 (and A241269).
Here we apply the transform to the Bernoulli numbers A164555(n)/A027642(n).
We take n*(0 followed by A164555(n)/A027642(n)).
Hence the autosequence of first kind
TB1(n) = 0, 1, 1, 1/2, 0, -1/6, 0, 1/6, 0, -3/10, 0, 5/6, O, -691/210,.. .
a(n) are the numerators.
The first seven rows of the differencece table of TB1(n) are
0,       1,    1,  1/2,   0, - 1/6,    0,      1/6,...
1,       0, -1/2, -1/2,  -1/6,  1/6,  1/6,    -1/6,... =A140351(n+1)/b(n+1)
-1,   -1/2,    0,  1/3,   1/3,    0, -1/3,   -2/15,...
1/2,   1/2,  1/3,    0,  -1/3, -1/3,  1/5,   11/15,...
0,    -1/6, -1/3, -1/3,     0, 8/15, 8/15,    -4/5,...
-1/6, -1/6,    0,  1/3,  8/15,    0, -4/3,    -4/3,...
0,     1/6,  1/3,  1/5, -8/15, -4/3,    0, 512/105,... .
First and second upper diagonals: 1, -1/2, 1/3, -1/3, 8/15, -4/3, 512/105,... .
Sum of the antidiagonals:
0, 1, 1, 0, -1/2, 0, 1/2, 0, -5/6, 0, 13/6, 0, -49/6, 0,... .
(Note that the same transform applied to the second fractional Euler numbers A198631(n)/A006519(n+1) yields the Genocchi numbers -A226158(n)).
This transform can be continued:
TB2(n) = n*(0 followed by TB1(n)) =
0, 0, 2, 3, 2, 0, -1, 0, 4/3, 0, -3, 0, 10, 0, -691/15, 0, 280, 0,...
is an autosequence of second kind.
TB3(n) = 0, 0, 0, 6, 12, 10, 0, -7, 0, 12, 0, -33, 0, 130, 0, 691, 0,...
is apparently an integer autosequence of the first kind.

Cf. A199969 (autosequence).

a(n) = 0 followed by (A050925(n) = 1, -1, 1, 0,... ) with 1 instead of -1.

a(2n) = A063524(n). a(2n+1) = A002427(n).

cp's OEIS Frontend

A342320 Integers k such that Euler(k, 1) is an integer multiple of Bernoulli(k + 1, 1).

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A227577 Square array read by antidiagonals, A(n,k) the numerators of the elements of the difference table of the Euler polynomials evaluated at x=1, for n>=0, k>=0.

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A271573 Numerator of (0 followed by A005126(n)= 2, 4, 7, ...)/2^n.

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A339058 a(n) = 4^n*Euler(n, 1/4)*2^(valuation_{2}(n + 1)).

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A242971 Alternate n+1, 2^n.

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A244213 Inverse binomial transform of -2 followed by A000032(n+1).

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A281500 Reduced denominators of f(n) = (n+1)/(2^(2+n)-2) with A026741(n+1) as numerators.

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A339058 a(n) = 4^nEuler(n, 1/4)2^(valuation_{2}(n + 1)).