A036968 -id:A036968 - cp's OEIS frontend

A083012 a(n) = Sum_{k=0..n-1} 8^kB(k)binomial(n,k) where B(k) is the k-th Bernoulli number.

0, 1, -7, 21, 49, -595, -1911, 39109, 165473, -4525731, -23883335, 805349237, 5097585297, -203564524787, -1503073984279, 69292329479205, 584713994953921, -30553447357629763, -290046835163027943, 16939595863125337813, 178676615255242261745

Offset: 0

Benoit Cloitre, May 31 2003

sign

Seiichi Manyama, Table of n, a(n) for n = 0..432

Cf. A001469.

Cf. A036968, A083007, A083008, A083009, A083010, A083011, A083013, A083014.

Range[0, 15]! CoefficientList[ Series[ 8x/(1 + Exp[x] + Exp[ 2x] + Exp[ 3x] + Exp[ 4x] + Exp[ 5x] + Exp[ 6x] + Exp[ 7x]), {x, 0, 15}], x] (* Robert G. Wilson v, Oct 26 2012 *)

a(n)=sum(k=0,n-1,8^k*binomial(n,k)*bernfrac(k))

Offset changed to 0 by Seiichi Manyama, Sep 28 2016

A083013 a(n) = Sum_{k=0..n-1} 9^kB(k)binomial(n,k) where B(k) is the k-th Bernoulli number.

Original entry on oeis.org

0, 1, -8, 28, 64, -980, -3104, 81172, 339328, -11878244, -61958240, 2674671076, 16735235392, -855605816468, -6245150369696, 368601472639540, 3074742020313856, -205700802920736452, -1930357641628367072, 144338957346266943364, 1505019970814899568320

Offset: 0

Benoit Cloitre, May 31 2003

sign

Seiichi Manyama, Table of n, a(n) for n = 0..424

Cf. A001469.

Cf. A036968, A083007, A083008, A083009, A083010, A083011, A083012, A083014.

Range[0, 15]! CoefficientList[ Series[ 9x/(1 + Exp[x] + Exp[ 2x] + Exp[ 3x] + Exp[ 4x] + Exp[ 5x] + Exp[ 6x] + Exp[ 7x] + Exp[ 8x]), {x, 0, 15}], x] (* Robert G. Wilson v, Oct 26 2012 *)
Table[Sum[9^k BernoulliB[k]Binomial[n,k],{k,0,n-1}],{n,0,20}] (* Harvey P. Dale, Apr 13 2016 *)

a(n)=sum(k=0,n-1,9^k*binomial(n,k)*bernfrac(k))

Offset changed to 0 by Seiichi Manyama, Sep 28 2016

A083014 a(n) = Sum_{k=0..n-1} 10^kB(k)binomial(n,k) where B(k) is the k-th Bernoulli number.

Original entry on oeis.org

0, 1, -9, 36, 81, -1524, -4779, 155316, 643761, -28041444, -145069299, 7794224196, 48371836041, -3078058903764, -22284938832219, 1637087002046676, 13545357290061921, -1127884947406124484, -10498665795419017539, 977073296798704710756

Offset: 0

Benoit Cloitre, May 31 2003

sign

Seiichi Manyama, Table of n, a(n) for n = 0..417
Ira M. Gessel, On the Almkvist-Meurman Theorem for Bernoulli Polynomials, Integers (2023) Vol. 23, #A14.

Cf. A001469.

Cf. A036968, A083007, A083008, A083009, A083010, A083011, A083012, A083013.

Range[0, 15]! CoefficientList[ Series[ 10x/(1 + Exp[x] + Exp[ 2x] + Exp[ 3x] + Exp[ 4x] + Exp[ 5x] + Exp[ 6x] + Exp[ 7x] + Exp[ 8x] + Exp[ 9x]), {x, 0, 15}], x] (* Robert G. Wilson v, Oct 26 2012 *)
Array[Sum[10^k*BernoulliB[k]*Binomial[#, k], {k, 0, # - 1}] &, 20, 0] (* Michael De Vlieger, Feb 14 2023 *)

a(n)=sum(k=0,n-1,10^k*binomial(n,k)*bernfrac(k))

E.g.f.: 10*x/(Sum_{i=0..9} exp(i*x)). - Alois P. Heinz, Sep 28 2016

Offset changed to 0 by Seiichi Manyama, Sep 28 2016

A162660 Triangle read by rows: coefficients of the complementary Swiss-Knife polynomials.

Original entry on oeis.org

0, 1, 0, 0, 2, 0, -2, 0, 3, 0, 0, -8, 0, 4, 0, 16, 0, -20, 0, 5, 0, 0, 96, 0, -40, 0, 6, 0, -272, 0, 336, 0, -70, 0, 7, 0, 0, -2176, 0, 896, 0, -112, 0, 8, 0, 7936, 0, -9792, 0, 2016, 0, -168, 0, 9, 0, 0, 79360, 0, -32640, 0, 4032, 0, -240, 0, 10, 0

Offset: 0

Peter Luschny, Jul 09 2009

Definition. V_n(x) = (skp(n, x+1) - skp(n, x-1))/2 where skp(n,x) are the Swiss-Knife polynomials A153641. - Peter Luschny, Jul 23 2012
Equivalently, let the polynomials V_n(x) (n>=0) defined by V_n(x) = Sum_{k=0..n} Sum_{v=0..k} (-1)^v*C(k,v)*L(k)*(x+v+1)^n; the sequence L(k) = -1 - H(k-1)*(-1)^floor((k-1)/4) / 2^floor(k/2) if k > 0 and L(0)=0; H(k) = 1 if k mod 4 <> 0, otherwise 0.
(1) V_n(0) = 2^n * Euler(n,1) for n > 0, A155585.
(2) V_n(1) = 1 - Euler(n).
(3) V_{n-1}(0) n / (4^n - 2^n) = B_n gives for n > 1 the Bernoulli numbers A027641/A027642.
(4) V_{n-1}(0) n (2/2^n-2)/(2^n-1) = G_n the Genocchi number A036968 for n > 1.
(5) V_n(1/2)2^{n} - 1 is a signed version of the generalized Euler (Springer) numbers, see A001586.
The Swiss-Knife polynomials (A153641) are complementary to the polynomials defined here. Adding both gives polynomials with e.g.f. exp(x*t)*(sech(t)+tanh(t)), the coefficients of which are a signed variant of A109449.
The Swiss-Knife polynomials as well as the complementary Swiss-Knife polynomials are closely related to the Bernoulli and Euler polynomials. Let F be a sequence and
P_{F}[n](x) = Sum_{k=0..n} Sum_{v=0..k} (-1)^v*C(k,v)*F(k)*(x+v+1)^n.
V_n(x) = P_{F}[n](x) with F(k)=L(k) defined above, are the Co-Swiss-Knife polynomials,
W_n(x) = P_{F}[n](x) with F(k)=c(k) the Chen sequence defined in A153641 are the Swiss-Knife polynomials.
B_n(x) = P_{F}[n](x-1) with F(k)=1/(k+1) are the Bernoulli polynomials,
E_n(x) = P_{F}[n](x-1) with F(k)=2^(-k) are the Euler polynomials.
The most striking formal difference between the Swiss-Knife-type polynomials and the Bernoulli-Euler type polynomials is: The SK-type polynomials have integer coefficients whereas the BE-type polynomials have rational coefficients.
Let R be the exponential Riordan array (exp(x)*sech(x), x) = P * A119879 = 2*P(I + P^2)^(-1) where P denotes Pascal's triangle A007318. Then T = R - I. - Peter Bala, Mar 07 2024

			Triangle begins:
  [0]    0;
  [1]    1,     0;
  [2]    0,     2,     0;
  [3]   -2,     0,     3,   0;
  [4]    0,    -8,     0,   4,    0;
  [5]   16,     0,   -20,   0,    5,    0;
  [6]    0,    96,     0, -40,    0,    6,    0;
  [7] -272,     0,   336,   0,  -70,    0,    7,  0;
  [8]    0, -2176,     0, 896,    0, -112,    0,  8,  0;
  [9] 7936,     0, -9792,   0, 2016,    0, -168,  0,  9,  0;

Ayse Yilmaz Ceylan and Yilmaz Simsek, Formulae for Generalization of Touchard Polynomials with Their Generating Functions, Symmetry (2025) Vol. 17, Issue 7, Art. No. 1126. See Eq. 28 and after.
Leonhard Euler (1735), De summis serierum reciprocarum, Opera Omnia I.14, E 41, 73-86; On the sums of series of reciprocals, arXiv:math/0506415 [math.HO], 2005-2008.
Peter Luschny, The Swiss-Knife polynomials.
Peter Luschny, Swiss-Knife Polynomials and Euler Numbers.
Wikipedia, Bernoulli number.
J. Worpitzky, Studien über die Bernoullischen und Eulerschen Zahlen, Journal für die reine und angewandte Mathematik, 94 (1883), 203-232.

V_n(k), n=0, 1, ..., k=0: A155585, k=1: A009832,
V_n(k), k=0, 1, ..., V_0: A000004, V_1: A000012, V_2: A005843, V_3: A100536.
Cf. A153641, A154341, A154342, A154343, A154344, A154345.

# Polynomials V_n(x):
V := proc(n,x) local k,pow; pow := (n,k) -> `if`(n=0 and k=0,1,n^k); add(binomial(n,k)*euler(k)*pow(x+1,n-k),k=0..n) - pow(x,n) end:
# Coefficients a(n):
seq(print(seq(coeff(n!*coeff(series(exp(x*t)*tanh(t),t,16),t,n),x,k),k=0..n)),n=0..8);

skp[n_, x_] := Sum[Binomial[n, k]*EulerE[k]*x^(n-k), {k, 0, n}]; v[n_, x_] := (skp[n, x+1]-skp[n, x-1])/2; t[n_, k_] := Coefficient[v[n, x], x, k]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 09 2014 *)

R = PolynomialRing(QQ, 'x')
@CachedFunction
def skp(n, x) : # Swiss-Knife polynomials A153641.
    if n == 0 : return 1
    return add(skp(k, 0)*binomial(n, k)*(x^(n-k)-(n+1)%2) for k in range(n)[::2])
def A162660(n,k) : return 0 if k > n else R((skp(n, x+1)-skp(n, x-1))/2)[k]
matrix(ZZ, 9, A162660) # Peter Luschny, Jul 23 2012

T(n, k) = [x^(n-k)](skp(n,x+1)-skp(n,x-1))/2 where skp(n,x) are the Swiss-Knife polynomials A153641. - Peter Luschny, Jul 23 2012
E.g.f. exp(x*t)*tanh(t) = 0*(t^0/0!) + 1*(t^1/1!) + (2*x)*(t^2/2!) + (3*x^2-2)*(t^3/3!) + ...
V_n(x) = -x^n + Sum_{k=0..n} C(n,k)*Euler(k)*(x+1)^(n-k).

A154344 Triangle read by rows. G(n, k) an additive decomposition of 2^n*G(n), G(n) the Genocchi numbers.

Original entry on oeis.org

1, 0, -2, 0, -3, 3, 0, -4, 12, 0, 0, -5, 35, 0, -30, 0, -6, 90, 0, -360, 180, 0, -7, 217, 0, -2730, 3150, -630, 0, -8, 504, 0, -16800, 33600, -15120, 0, 0, -9, 1143, 0, -91854, 283500, -215460, 0, 22680, 0, -10, 2550, 0, -466200, 2085300, -2381400, 0, 907200, -226800

Offset: 0

Peter Luschny, Jan 07 2009

The Swiss-Knife polynomials A153641 can be understood as a sum of polynomials. Evaluated at x=-1 multiplied by n+1 this results in a decomposition of 2^n times the Genocchi numbers A036968.

			Triangle begins:
  1,
  0, -2,
  0, -3,    3,
  0, -4,   12, 0,
  0, -5,   35, 0,    -30,
  0, -6,   90, 0,   -360,    180,
  0, -7,  217, 0,  -2730,   3150,    -630,
  0, -8,  504, 0, -16800,  33600,  -15120, 0,
  0, -9, 1143, 0, -91854, 283500, -215460, 0, 22680,
  ...

G. C. Greubel, Table of n, a(n) for n = 0..1274
Ayse Yilmaz Ceylan and Yilmaz Simsek, Formulae for Generalization of Touchard Polynomials with Their Generating Functions, Symmetry (2025) Vol. 17, Issue 7, Art. No. 1126. See Eq. 28 and after.
Peter Luschny, The Swiss-Knife polynomials.

Cf. A036968, A153641, A154341, A154342, A154343, A154345.

G := 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)*(n+1)*c(k)*v^n, v=0..k) end: seq(print(seq(G(n, k), k=0..n)), n=0..8);

g[n_, k_] := Module[{v, c, pow}, pow[a_, b_] := If[ a == 0 && b == 0, 1, a^b]; c[m_] := If[ Mod[m+1, 4] == 0 , 0 , 1/((-1)^Quotient[m+1, 4]*2^Quotient[m, 2])]; Sum[(-1)^v*Binomial[k, v]*(n+1)*c[k]*pow[v, n], {v, 0, k}]]; Table[g[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 23 2013, translated from Maple *)

Let c(k) = ((-1)^floor(k/4) / 2^floor(k/2)) * [4 not div k+1] (Iverson notation).
G(n,k) = Sum_{v=0..k} (-1)^v*binomial(k,v)*(n+1)*c(k)*v^n.
A036968(n) = (1/2^n)*Sum_{k=0..n} G(n,k).

A347599 Irregular table read by rows, T(n, k) is the rank of the k-th Genocchi permutation of {1,...,n}, permutations sorted in lexicographical order. If no Genocchi permutation of {1,...,n} exists, then T(n, 1) = 0 by convention.

Original entry on oeis.org

1, 0, 5, 0, 67, 91, 92, 0, 1897, 2017, 2018, 2617, 2619, 2737, 2738, 2739, 2740, 3457, 3458, 3459, 3460, 4177, 4178, 4179, 4180, 0, 99241, 99961, 99962, 104281, 104283, 105001, 105002, 105003, 105004, 110041, 110042, 110043, 110044, 115081, 115082, 115083

Offset: 1

Peter Luschny, Sep 08 2021

Let M be the n X n matrix with M(j, k) = floor((2*j - k ) / n). A Genocchi permutation of order n is a permutation sigma of {1,...,n} if Product_{k=1..n} M(k, sigma(k)) does not vanish.
Let P(n) denote the number of Genocchi permutations of order n. Zhi-Wei Sun conjectured, using permanents, that P(n - 1) = G(n), where G(n) are the Genocchi numbers A036968. From the well-known relation between Genocchi and Bernoulli numbers this implies, assuming the conjecture:
  Bernoulli(n) = P(n - 1) / ((-1)^floor(n/2)*(2^(n + 2) - 2)) for n >= 2.
The related sequence A347600 lists Seidel permutations.

			Table starts:
[1] 1;
[2] 0;
[3] 5;
[4] 0;
[5] 67, 91, 92;
[6] 0;
[7] 1897, 2017, 2018, 2617, 2619, 2737, 2738, 2739, 2740, 3457, 3458, 3459, 3460, 4177, 4178, 4179, 4180;
.
The 17 permutations corresponding to the ranks are for n = 7:
1897 -> [3571246]; 2017 -> [3671245]; 2018 -> [3671254]; 2617 -> [4571236];
2619 -> [4571326]; 2737 -> [4671235]; 2738 -> [4671253]; 2739 -> [4671325];
2740 -> [4671352]; 3457 -> [5671234]; 3458 -> [5671243]; 3459 -> [5671324];
3460 -> [5671342]; 4177 -> [6571234]; 4178 -> [6571243]; 4179 -> [6571324];
4180 -> [6571342].
.
17 / (-510) = -1/30 = Bernoulli(8).

Zhi-Wei Sun, A novel identity connecting permanents to Bernoulli numbers, MathOverflow 2021-09-07.

Cf. A036968, A226158, A027641/A027642, A347600.

using Combinatorics
function GenocchiPermutations(n)
    f(m) = m >= n ? 1 : m < 0 ? -1 : 0
    Mat(n) = [[f(2*j - k) for k in 1:n] for j in 1:n]
    M = Mat(n); P = permutations(1:n); R = Int64[]
    S, rank = 0, 1
    for p in P
        m = prod(M[k][p[k]] for k in 1:n)
        if m != 0
            S += m
            push!(R, rank)
        end
        rank += 1
    end
    # println(n, "  ", S, "  ", S // (2^(n + 2) - 2)) # Bernoulli number
    return R
end
for n in 1:11 println(GenocchiPermutations(n)) end

A347600 Irregular table read by rows, T(n, k) is the rank of the k-th Seidel permutation of {1,...,n}, permutations sorted in lexicographical order.

Original entry on oeis.org

2, 11, 17, 187, 211, 307, 331, 451, 452, 571, 572, 6937, 7057, 7657, 7777, 8497, 8498, 9217, 9218, 11977, 12097, 12697, 12817, 13537, 13538, 14257, 14258, 17737, 17739, 17857, 17859, 18577, 18578, 18579, 18580, 19297, 19298, 19299, 19300, 22777, 22779, 22897

Offset: 1

Peter Luschny, Sep 08 2021

Let M be the 2n X 2n matrix with M(j, k) = floor((2*j - k - 1) / 2*n). A Seidel permutation of order n is a permutation sigma of {1,...,2n} if Product_{k=1..2n} M(k, sigma(k)) does not vanish.
Let P(n) denote the number of Seidel permutations of order n. We conjecture that P(n) = A005439(n). This conjecture was inspired by the conjecture of Zhi-Wei Sun in A036968. The name 'Seidel permutations' follows a comment of Don Knuth: "The earliest known reference for these numbers (A005439) is Seidel ...."
The related sequence A347599 lists Genocchi permutations.

			Table starts:
[1] 2;
[2] 11, 17;
[3] 187, 211, 307, 331, 451, 452, 571, 572.
.
The 8 permutations corresponding to the ranks are for n = 3:
187 -> [246135]; 211 -> [256134]; 307 -> [346125]; 331 -> [356124];
451 -> [456123]; 452 -> [456132]; 571 -> [546123]; 572 -> [546132].

Cf. A005439, A036968, A347599.

function SeidelPermutations(n)
    f(m) = m >= 2n ? 1 : m < 0 ? -1 : 0
    Mat(n) = [[f(2*j - k - 1) for k in 1:2n] for j in 1:2n]
    M = Mat(n); P = permutations(1:2n); R = Int64[]
    S, rank = 0, 1
    for p in P
        m = prod(M[k][p[k]] for k in 1:2n)
        if m != 0
            S += m
            push!(R, rank)
        end
        rank += 1
    end
    # println(n, " -> ", (-1)^n*S)
    return R
end
for n in 1:5 println(SeidelPermutations(n)) end

A224783 Denominator of Bernoulli(n,1/2) - Bernoulli(n,0).

Original entry on oeis.org

1, 2, 4, 1, 16, 1, 64, 1, 256, 1, 1024, 1, 4096, 1, 16384, 1, 65536, 1, 262144, 1, 1048576, 1, 4194304, 1, 16777216, 1, 67108864, 1, 268435456, 1, 1073741824, 1, 4294967296, 1, 17179869184, 1, 68719476736, 1, 274877906944, 1, 1099511627776

Offset: 0

Paul Curtz, Apr 17 2013

See A157779 and A157780 for values of Bernoulli(n,1/2), and A027641 and A027642 for values of Bernoulli(n,0).
B(n,1/2) - B(n,0) = 0, 1/2, -1/4, 0, 1/16, 0, -3/64, 0, 17/256, 0, -155/1024, 0, 2073/4096, 0, -38227/16384,... for n>=0.
The sequence of numerators is 0, 1, -1, 0, 1, 0, -3, 0, 17, 0, -155, 0, 2073, 0, -38227, 0, 929569, 0, -28820619, 0, 1109652905,...and appears to contain a mix of A001469 and A036968.

			a(0) = 1-1, a(1) = 0+1/2, a(2) = -1/12-1/6=-1/4.

Vincenzo Librandi, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-4).

Cf. A001469, A027641, A027642, A036968, A059222, A157779, A157780.

A224783 := proc(n)
    bernoulli(n,1/2)-bernoulli(n) ;
    denom(%) ;
end proc: # R. J. Mathar, Apr 25 2013

Table[Denominator[BernoulliB[n, 1/2] - BernoulliB[n, 0]], {n, 0, 50}] (* Vincenzo Librandi, Mar 19 2014 *)

Vec((4*x^5-9*x^3-x^2+2*x+1)/((x-1)*(x+1)*(2*x-1)*(2*x+1)) + O(x^100)) \\ Colin Barker, Mar 20 2014

a(n) = A059222(n+1) if n <> 1.
From Colin Barker, Mar 19 2014: (Start)
G.f.: (4*x^5-9*x^3-x^2+2*x+1) / ((x-1)*(x+1)*(2*x-1)*(2*x+1)).
a(n) = 5*a(n-2)-4*a(n-4) for n>5.
a(n) = (1+(-2)^n-(-1)^n+2^n)/2 for n>1. (End).

A225825 a(2n)=A001896(n). a(2n+1)=(-1)^n*A110501(n+1).

Original entry on oeis.org

1, 1, -1, -1, 7, 3, -31, -17, 127, 155, -2555, -2073, 1414477, 38227, -57337, -929569, 118518239, 28820619, -5749691557, -1109652905, 91546277357, 51943281731, -1792042792463, -2905151042481, 1982765468311237, 191329672483963, -286994504449393, -14655626154768697, 3187598676787461083, 1291885088448017715, -4625594554880206790555

Offset: 0

Paul Curtz, Jul 30 2013

sign

a(n) is the numerators of numbers derived from Bernoulli and Genocchi numbers. The denominators b(n) are the Clausen numbers A141056.
The numbers are
BERGEN(n) = 1, 1/2, -1/6, -1/2, 7/30, 3/2, -31/42, -17/2, 127/30, 155/2,..
Difference table:
1,       1/2,     -1/6,     -1/2,      7/30,        3/2,    -31/42,...
-1/2,   -2/3,     -1/3,    11/15,     19/15,     -47/21,   -163/21,...
-1/6,    1/3,    16/15,     8/15,  -368/105,    -116/21,  2152/105,...
1/2,   11/15,    -8/15, -424/105,  -212/105,   2732/105,  4204/105,...
7/30, -19/15, -368/195,  212/105,  2944/105,   1472/105,...
-3/2, -47/21,   116/21, 2732/105, -1472/105, -70240/231, -35120/231,... .
a(n) is an autosequence. Its inverse binomial transform is the sequence signed. Its main diagonal is the double of the first upper diagonal.
a(n) is divisible by A051716(n+1).
Denominators of the main diagonal: A181131(n). Checked by Jean-François Alcover for the first 25 terms.
The numerators of the main diagonal:
1, -2, 16, -424, 2944, -70240, 70873856, -212648576, 98650550272,...
(thanks to Jean-François Alcover) are divisible by 2^n.

Cf. A083420.

A225825 := proc(n)
    local nhalf ;
    nhalf := floor(n/2) ;
    if type(n,'even') then
        A001896(nhalf) ;
    else
        (-1)^nhalf*A110501(nhalf+1) ;
    end if;
end proc; # R. J. Mathar, Oct 28 2013

a[0] = 1; a[n_] := Numerator[BernoulliB[n, 1/2] - (n+1)*EulerE[n, 0]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Aug 01 2013 *)

c(n)=(0 followed by -A036968(n+1)) = 0, 1, 0, -1, 0, 3,... .

a(n) = A157779(n) + c(n).

More terms from Jean-François Alcover, Aug 01 2013

Definition corrected by R. J. Mathar, Oct 28 2013

A179077 a(n) is the residue ((2^p - 2)/p) mod p, where p is the n-th prime.

Original entry on oeis.org

1, 2, 1, 4, 10, 6, 9, 6, 11, 2, 12, 2, 5, 7, 41, 19, 16, 11, 20, 4, 39, 38, 13, 12, 17, 83, 15, 26, 25, 53, 36, 34, 106, 60, 43, 112, 7, 134, 94, 6, 100, 115, 100, 15, 153, 71, 7, 155, 175, 136, 14, 52, 43, 243, 193, 256, 251, 218, 140, 148, 116, 156, 281, 39, 240, 33, 278

Offset: 1

Artur Jasinski, Jun 28 2010

nonn

a(n) = 0 where n=183 (p=1093) and n=490 (p=3511).
From Felix Fröhlich, Sep 13 2019: (Start)
Conjecture: a(n) is the residue A036968(p-1) (mod p) for p = prime(n).
If the above conjecture is true, then a(n) = 0 if and only if p is a Wieferich prime (A001220) (cf. Hu et al., 2019, section 1.3). (End)

Robert Israel, Table of n, a(n) for n = 1..10000
Su Hu, Min-Soo Kim, Pieter Moree and Min Sha, Irregular primes with respect to Genocchi numbers and Artin's primitive root conjecture, arXiv:1809.08431 [math.NT], 2019; Journal of Number Theory 205 (2019), 59-80, DOI:10.1016/j.jnt.2019.03.012.

Cf. A001220, A036968, A377655.

f:= p -> (2&^p-2 mod p^2)/p:
seq(f(ithprime(i)),i=1..100);# Robert Israel, Nov 03 2024

aa = {}; Do[AppendTo[aa, Mod[(2^Prime[n] - 2)/Prime[n], Prime[n]]], {n, 1, 100}]; aa

a(n) = my(p=prime(n)); lift(Mod(((2^p-2)/p), p)) \\ Felix Fröhlich, Sep 13 2019

cp's OEIS Frontend

A083012 a(n) = Sum_{k=0..n-1} 8^k*B(k)*binomial(n,k) where B(k) is the k-th Bernoulli number.

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A083013 a(n) = Sum_{k=0..n-1} 9^k*B(k)*binomial(n,k) where B(k) is the k-th Bernoulli number.

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A083014 a(n) = Sum_{k=0..n-1} 10^k*B(k)*binomial(n,k) where B(k) is the k-th Bernoulli number.

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A162660 Triangle read by rows: coefficients of the complementary Swiss-Knife polynomials.

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A154344 Triangle read by rows. G(n, k) an additive decomposition of 2^n*G(n), G(n) the Genocchi numbers.

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A347599 Irregular table read by rows, T(n, k) is the rank of the k-th Genocchi permutation of {1,...,n}, permutations sorted in lexicographical order. If no Genocchi permutation of {1,...,n} exists, then T(n, 1) = 0 by convention.

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Julia

A347600 Irregular table read by rows, T(n, k) is the rank of the k-th Seidel permutation of {1,...,n}, permutations sorted in lexicographical order.

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Julia

A224783 Denominator of Bernoulli(n,1/2) - Bernoulli(n,0).

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A083012 a(n) = Sum_{k=0..n-1} 8^kB(k)binomial(n,k) where B(k) is the k-th Bernoulli number.

A083013 a(n) = Sum_{k=0..n-1} 9^kB(k)binomial(n,k) where B(k) is the k-th Bernoulli number.

A083014 a(n) = Sum_{k=0..n-1} 10^kB(k)binomial(n,k) where B(k) is the k-th Bernoulli number.