A239315 -id:A239315 - cp's OEIS frontend

A168426 Square array of denominators of a truncated array of Bernoulli twin numbers (A168516), read by antidiagonals.

3, 6, 6, 30, 15, 30, 30, 15, 15, 30, 42, 105, 105, 105, 42, 42, 21, 105, 105, 21, 42, 30, 105, 105, 105, 105, 105, 30, 30, 15, 105, 105, 105, 105, 15, 30, 66, 165, 165, 1155, 231, 1155, 165, 165, 66, 66, 33, 165, 165, 231, 231, 165, 165, 33, 66, 2730, 15015, 15015, 15015, 15015, 15015, 15015, 15015

Offset: 0

Paul Curtz, Nov 25 2009

Entries are multiples of 3.
The sequence of fractions A051716()/A051717() is a sequence of first differences of A164555()/A027642().
It can be observed (see the difference array in A190339) that A168516/A168426 is a sequence of autosequences of the second kind. - Paul Curtz, Dec 21 2016

Cf. A085738, A085737, A168516, A190339, A240581/A239315.

max = 11; c[0] = 1; c[n_?EvenQ] := BernoulliB[n] + BernoulliB[n-1]; c[n_?OddQ] := -BernoulliB[n] - BernoulliB[n-1]; cc = Table[c[n], {n, 0, max+1}]; diff = Drop[#, 2]& /@ Table[ Differences[cc, n], {n, 0, max-1}]; Flatten[ Table[ diff[[n-k+1, k]], {n, 1, max}, {k, 1, n}]] // Denominator (* Jean-François Alcover, Aug 09 2012 *)

More terms from R. J. Mathar, Jul 10 2011

A240581 Array read by antidiagonals: numerators of the core of the classical Bernoulli numbers.

Original entry on oeis.org

2, -1, 1, -1, -8, -1, 1, 4, -4, -1, -1, 4, 8, 4, -1, -1, -8, -4, 4, 8, 1, 7, -4, -116, -32, -116, -4, 7, 5, 32, 28, 16, -16, -28, -32, -5, -2663, -388, 2524, 5072, 6112, 5072, 2524, -388, -2663, -691, -10264, -10652, -8128, -3056, 3056, 8128, 10652, 10264, 691

Offset: 0

Paul Curtz, Apr 08 2014

Sum of antidiagonals: 2/15, 0, -2/21, 0, 2/15, 0, -10/33, 0, 1382/1365,... =-4*A164555(n+4)/A027642(n+4).

			As a triangle:
2,
-1,  1,
-1, -8, -1,
1,   4, -4, -1,
-1,  4,  8,  4, -1,
etc.

Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187.

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

Cf. A239315 (denominators).

DifferenceTableBernoulli := proc(n) local A,m,k; A := array(0..n,0..n);
# pritty print
for m from 0 to n do for k from 0 to n do A[m, k] := '~' od od;
# compute elements
for m from 0 to n do A[m,0] := bernoulli(m, 1);
   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;
convert(A, matrix) end:
A := DifferenceTableBernoulli(13); L := NULL;
for n from 0 to 9 do for k from 0 to n do
   L := L, numer(A[3+k, 3+n-k]) od od;
L; # Peter Luschny, Apr 12 2014

max = 13; tb = Table[BernoulliB[n], {n, 0, max}]; td = Table[Differences[tb, n][[3 ;; -1]], {n, 2, max - 1}]; Table[td[[n - k + 1, k]] // Numerator, {n, 1, max - 3}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 11 2014 *)

a(9) corrected by Wesley Ivan Hurt, Apr 08 2014

a(4) corrected by Jean-François Alcover, Apr 11 2014

A239977 a(n) = -Sum_{k=0..n} binomial(n, k)*A226158(k).

Original entry on oeis.org

0, 1, 3, 6, 9, 10, 9, 14, 33, 18, -135, 22, 2097, 26, -38199, 30, 929601, 34, -28820583, 38, 1109652945, 42, -51943281687, 46, 2905151042529, 50, -191329672483911, 54, 14655626154768753, 58, -1291885088448017655, 62, 129848163681107302017, 66

Offset: 0

Paul Curtz, Mar 30 2014

Let T(n, k) denote the difference table of a(n).
(-1)^(k+1)*T(3, 3+k) = T(k+3, 3) for k >= 0.
Without the first two rows and the first two columns we have the core of the Genocchi numbers, like A240581(n)/A239315(n) for the Bernoulli numbers. See A226158(n).
   0,   1,   3,   6,   9,  10, ...
   1,   2,   3,   3,   1,  -1, ...
   1,   1,   0,  -2,  -2,   6, ...
   0,  -1,  -2,   0,   8,   8, ...
  -1,  -1,   2,   8,   0, -56, ...
   0,   3,   6,  -8, -56,   0, ...
The definition reflects the identity 2*((1-2^n)*B(n,1) + n) =
2*Sum_{k=0..n} C(n,k)*(2^k-1)*B(k,1) where B(n,x) denotes the Bernoulli polynomials. - Peter Luschny, Apr 16 2014

B. Sury, The value of Bernoulli Polynomials at rational numbers, Bull. London Math. Soc. 25 (1993), 327-29.

Cf. A083007.

[0,1] cat [2*((1 - 2^n)*Bernoulli(n) + n): n in [2..40]]; // Vincenzo Librandi, Mar 03 2015

A239977 := n -> 2*((1-2^n)*bernoulli(n,1) + n):
seq(A239977(n), n=0..33); # Peter Luschny, Mar 08 2015

a[n_] := (EulerE[n-1, 0]+2)*n; a[0] = 0; a[1] = 1; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 02 2014 *)

a(n) = 2*n + A226158(n).
a(2n) is divisible by 3.
a(2n+1) = A133653(n).
a(n) = 2*((1 - 2^n)*B(n, 1) + n), B(n, x) the Bernoulli polynomial. - Peter Luschny, Apr 16 2014
From Peter Bala, Mar 02 2015: (Start)
a(n) = (-2)^n * ( B(n,-1/2) - B(n,0) ), where B(n,x) denotes the n-th Bernoulli polynomial. More generally, for any nonzero integer k, k^n*( B(n,1/k) - B(n,0) ) is an integer for n >= 0. Cf. A083007.
a(0) = 0 and for n >= 1, a(n) = 1 - 1/(n + 1)*Sum_{k = 1..n-1} (-2)^(n-k)*binomial(n+1,k)*a(k).
E.g.f.: 2*x*exp(2*x)/(1 + exp(x)) = x + 3*x^2/2! + 6*x^3/3! + .... (End)
a(n) = Sum_{k=0..n-1} 2^k*binomial(n, k)*Bernoulli(k, 1). - Peter Luschny, Aug 17 2021

A256675 Denominators of the inverse binomial transform of Bernoulli(n+2).

Original entry on oeis.org

6, 6, 15, 15, 105, 21, 105, 15, 165, 33, 15015, 1365, 1365, 3, 255, 255, 33915, 399, 21945, 165, 3795, 69, 31395, 1365, 1365, 3, 435, 435, 1038345, 7161, 608685, 255, 255, 3, 959595, 959595, 959595, 3, 6765, 6765, 2036265, 903, 103845, 345, 16215, 141, 1090635

Offset: 0

Paul Curtz, Apr 07 2015

nonn

Difference table of B(n+2):
1/6,        0,  -1/30,     0,   1/42,     0, -1/30, ...
-1/6,   -1/30,   1/30,  1/42,  -1/42, -1/30,   ...
2/15,    1/15, -1/105, -1/21, -1/105,   ...
-1/15, -8/105, -4/105, 4/105,    ...
-1/105, 4/105,  8/105,   ...
1/21,   4/105,    ...
-1/105,   ...
...
a(n) is the denominator of the n-th term of the first column.
a(n+2) is the denominator of the n-th term of the third row.
See A239315(n), which is the table without the first two rows.
Inverse binomial transform: 1/6, -1/6, 2/15, -1/15, -1/105, 1/21, -1/105, -1/15, 7/165, 5/33, -2663/15015, ... .

Cf. A190339, A239315, A029765, A001897, A141459, A172087, A168516, A177735, A256595.

max = 42; bb = Table[BernoulliB[n+2], {n, 0, max}]; dd = Table[Differences[bb, n], {n, 0, max}]; dd[[All, 1]] // Denominator (* Jean-François Alcover, Apr 09 2015 *)

lista(nn) = {A = vector(nn, n, bernfrac(n+1)); for (i=1, #A-1, for(j=0,i-1,A[i+1]-=binomial(i,j)*A[j+1])); for (i=1, #A, print1(denominator(A[i]), ", "));} \\ Michel Marcus, Apr 08 2015

a(2n) = A029765(n).
a(2n+3) = A001897(n+2).
a(2n)/a(2n+1) = A177735(n).
a(2n+4)/a(2n+3) = A177735(n+3).

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.

cp's OEIS Frontend

A168426 Square array of denominators of a truncated array of Bernoulli twin numbers (A168516), read by antidiagonals.

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A240581 Array read by antidiagonals: numerators of the core of the classical Bernoulli numbers.

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A239977 a(n) = -Sum_{k=0..n} binomial(n, k)*A226158(k).

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A256675 Denominators of the inverse binomial transform of Bernoulli(n+2).

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A278331 Shifted sequence of second differences of Genocchi numbers.

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