cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

Views

Author

Paul Curtz, Nov 25 2009

Keywords

Comments

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

Crossrefs

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

Programs

  • Mathematica
    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 *)

Extensions

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

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

Views

Author

Paul Curtz, Mar 30 2014

Keywords

Comments

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

Links

Crossrefs

Cf. A083007.

Programs

  • Magma
    [0,1] cat [2*((1 - 2^n)*Bernoulli(n) + n): n in [2..40]]; // Vincenzo Librandi, Mar 03 2015
  • Maple
    A239977 := n -> 2*((1-2^n)*bernoulli(n,1) + n):
seq(A239977(n), n=0..33); # Peter Luschny, Mar 08 2015
  • Mathematica
    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 *)

Formula

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

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

Views

Author

Jean-François Alcover and Paul Curtz, Nov 18 2016

Keywords

Comments

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.

Links

Crossrefs

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

Programs

  • Mathematica
    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}]

Formula

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.
Showing 1-3 of 3 results.

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