A005439 -id:A005439 - cp's OEIS frontend

A229023 Numerators of the main diagonal of A225825 difference table, a sequence linked to Bernoulli, Genocchi and Clausen numbers.

1, -2, 16, -424, 2944, -70240, 70873856, -212648576, 98650550272, -90228445612544, 19078660567134208, -2034677178643867648, 123160010212358914048, -19182197131374977024, 228111332170536254898176, -51166426240975948419354886144

Offset: 0

Jean-François Alcover and Paul Curtz, Sep 11 2013

a(n) is divisible by 2^n and congruent to 1, 2, 4, 5, 7 or 8 mod 9.

			1, -2/3, 16/15, -424/105, 2944/105, -70240/231, 70873856/15015, ...

Cf. A181131 (denominators), A225825, A110501 (Genocchi numbers), A141056 (Clausen numbers), A212196 (Bernoulli medians), A005439 (Genocchi medians).

nmax = 30; Clausen[n_] := Times @@ Select[Divisors[n] + 1, PrimeQ]; t = Join[{1}, Table[Numerator[BernoulliB[n, 1/2] - (n + 1)*EulerE[n, 0]]/Clausen[n], {n, 1, nmax}]]; dt = Table[Differences[t, n], {n, 0, nmax}]; Diagonal[dt] // Numerator

A240677 a(n) = 6Zeta(1-n)n*(2^n-1) - Zeta(-n)(n+1)(2^(n+2)-2), for n = 0 the limit is understood.

Original entry on oeis.org

1, -2, -3, -1, 3, 3, -9, -17, 51, 155, -465, -2073, 6219, 38227, -114681, -929569, 2788707, 28820619, -86461857, -1109652905, 3328958715, 51943281731, -155829845193, -2905151042481, 8715453127443

Offset: 0

Paul Curtz, Apr 10 2014

sign

G2(m, n), difference table of a(n):
1,   -2, -3, -1,   3,   3, -9, -17, 51,...
-3,  -1,  2,  4,   0, -12, -8,  68,...
2,    3,  2, -4, -12,   4, 76,...
1,   -1, -6, -8,  16,  72,...
-2,  -5, -2, 24,  56,...
-3,   3, 26, 32,...
6,   23,  6,...
17, -17,...
-34,...
etc.
The main diagonal G2(n,n) = 1, -1, 2, -8,... is essentially a signed version of A005439.
The first upper diagonal is the main diagonal multiplied by -2. G2(n, n+1) = -2*G2(n, n).
G2(m, n) = G2(m, n-1) + G2(m+1, n-1).
a(n) = (-1)^n*b(n) of A240485(n).
Inverse binomial transform: (-1)^n*A240485(n).
a(n) and A240485(n) are reciprocal. Like for instance (-1)^n and 2^n.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A240485.

A240677 := n -> `if`(n=0, 1, 6*Zeta(1-n)*n*(2^n-1) - Zeta(-n)*(n+1)*(2^(n+2)-2)); seq(A240677(n), n=0..24); # Peter Luschny, Apr 11 2014

g[0] = 0; g[1] = -1; g[n_] := n*EulerE[n - 1, 0]; a[n_] := 3*g[n] - g[n + 1]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 10 2014 *)

x = 'x+O('x^66);
A = -2*exp(x)*(2*x+exp(x)*(3*x-1)-1)/(exp(x)+1)^2;
Vec( serlaplace(A) )  /* Peter Luschny, Apr 10 2014 */

a(n) = 3*A226158(n) - A226158(n+1).
a(n+3) = -A001469(n+1).
a(2n+4) = -3*a(2n+3).
a(n) = A240485(n) + 5*A226158(n).
E.g.f.: -2*exp(x)*(2*x+exp(x)*(3*x-1)-1)/(exp(x)+1)^2. - Peter Luschny, Apr 10 2014

A278331 Shifted sequence of second differences of Genocchi numbers.

Original entry on oeis.org

0, -2, -2, 6, 14, -34, -138, 310, 1918, -4146, -36154, 76454, 891342, -1859138, -27891050, 57641238, 1080832286, -2219305810, -50833628826, 103886563462, 2853207760750, -5810302084962, -188424521441482, 382659344967926, 14464296482284734, -29311252309537394, -1277229462293249018

Offset: 0

Jean-François Alcover and Paul Curtz, Nov 18 2016

sign

This is an autosequence of the first kind (array of successive differences shows typical zero diagonal).
Last digits are apparently of period 20.
From A226158(n) for the continuity of autosequences of the first kind.
b(n) = 0, 1, -1, 0, 1, 0, -3, 0, 17, ... = A226158(n) with 1 as second term instead of -1.
c(n) = 0, 0, -1, 0, 1, 0, -3, 0, 17, ... = A226158(n) with 0 as second term instead of -1.
Respective difference tables:
0, -1, -1,  0,  1,  0, -3,   0,   17, ...
-1, 0,  1,  1, -1, -3 , 3,  17,  -17, ...
1,  1,  0, -2, -2,  6, 14, -34, -138, ...
etc,
0,  1, -1,  0,  1,  0, -3,   0,   17, ... = 0 followed by A036968(n+1)
1, -2,  1,  1, -1, -3,  3,  17,  -17, ...
-3, 3,  0, -2, -2,  6, 14, -34, -138, ...
etc,
0,  0, -1,  0,  1,  0, -3,   0,   17, ...
0, -1,  1,  1, -1, -3,  3,  17,  -17, ...
-1, 2,  0, -2, -2,  6, 14, -34, -138, ...
etc.
Since it is in the three tables, a(n) is the core of the Genocchi numbers.

Eric Weisstein's MathWorld, Genocchi Number.
Wikipedia, Genocchi number

Cf. A001469, A014781, A036968, A005439 (a(n) second and third diagonals), A164555/A027642, A209308, A226158, A240581(n)/A239315(n) (core of Bernoulli numbers).

g[0] = 0; g[1] = -1; g[n_] := n*EulerE[n-1, 0]; G = Table[g[n], {n, 0, 30}]; Drop[Differences[G, 2], 2]
(* or, from Seidel's triangle A014781: *)
max = 26; T[1, 1] = 1; T[n_, k_] /; 1 <= k <= (n + 1)/2 := T[n, k] = If[EvenQ[n], Sum[T[n - 1, i], {i, k, max}], Sum[T[n - 1, i], {i, 1, k}]]; T[, ] = 0; a[n_] := With[{k = Floor[(n - 1)/2] + 1}, (-1)^k*T[n + 3, k]]; Table[a[n], {n, 0, max}]

a(n) = (n+2)*E(n+1, 0) - 2*(n+3)*E(n+2, 0) + (n+4)*E(n+3, 0), where E(n,x) is the n-th Euler polynomial.

a(n) = -2*(2^(n+2)-1)*B(n+2) + 4*(2^(n+3)-1)*B(n+3) - 2*(2^(n+4)-1)*B(n+4), where B(n) is the n-th Bernoulli number.

A353921 a(n) = n if n < 4, otherwise floor(abs(z(n))) where z(n) = (2^(2n + 1/2) - 1)(4n + 1)zeta(1/2 - 2*n).

Original entry on oeis.org

0, 1, 2, 3, 20, 202, 2953, 58574, 1517830, 49788988, 2016610506, 98842394546, 5766037456673, 394787840828770, 31350291022336674, 2858009622374873775, 296454369597967332107, 34715387135986234970960, 4557676382296459474148951, 666708107998151285537770827

Offset: 0

Peter Luschny, May 14 2022

nonn

a(n) gives an integer valued definition of what may be called a 'Genocchi half integer', i.e. it tries to give the expression 'G(n + 1/2)' a meaning, where G(n) = A110501(n) are the Genocchi numbers. Consider also the sequence of Genocchi median numbers A005439.

Cf. A005439, A110501.

z := n -> (2^(2*n + 1/2) - 1)*(4*n + 1)*Zeta(1/2 - 2*n):
a := n -> ifelse(n < 4, n, floor(abs(z(n)))):
seq(floor(evalf(a(n))), n = 0..19);

A005439(n-1) <= a(n) <= A005439(n).
A110501(n) <= a(n) <= A110501(n+1).
a(n) ~ ((2*n)/(exp(1)*Pi))^(2*n)*(11/6 + 8*n - 23/(576*n)).

A362111 Related to shifted Genocchi medians.

Original entry on oeis.org

1, 1, 5, 41, 493, 8161, 178469, 4998905, 174914077, 7487810257, 385307632469, 23477954308841, 1672313866643917, 137703723643494721, 12981787835378567045, 1389285520249200892889, 167515121086596348291709, 22605814306270350503723185, 3393854148421455726168715445

Offset: 0

N. J. A. Sloane, Apr 14 2023

nonn

Bishal Deb and Alan D. Sokal, Classical continued fractions for some multivariate polynomials generalizing the Genocchi and median Genocchi numbers, arXiv:2212.07232 [math.CO], 2022. See (2.44).

Cf. A005439.

Rest@CoefficientList[Fold[#2/(1-#1)&,O[t],t Reverse@Table[Ceiling[n/2]^2,{n,20}]],t] (* Andrei Zabolotskii, Jul 09 2025 *)

Terms a(10) onwards from Andrei Zabolotskii, Jul 09 2025

cp's OEIS Frontend

A229023 Numerators of the main diagonal of A225825 difference table, a sequence linked to Bernoulli, Genocchi and Clausen numbers.

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A240677 a(n) = 6*Zeta(1-n)*n*(2^n-1) - Zeta(-n)*(n+1)*(2^(n+2)-2), for n = 0 the limit is understood.

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A278331 Shifted sequence of second differences of Genocchi numbers.

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A353921 a(n) = n if n < 4, otherwise floor(abs(z(n))) where z(n) = (2^(2*n + 1/2) - 1)*(4*n + 1)*zeta(1/2 - 2*n).

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A362111 Related to shifted Genocchi medians.

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A240677 a(n) = 6Zeta(1-n)n*(2^n-1) - Zeta(-n)(n+1)(2^(n+2)-2), for n = 0 the limit is understood.

A353921 a(n) = n if n < 4, otherwise floor(abs(z(n))) where z(n) = (2^(2n + 1/2) - 1)(4n + 1)zeta(1/2 - 2*n).