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.

Showing 1-3 of 3 results.

A240485 a(n) = -Zeta(1-n)*n*(2^(n+1) - 4) - Zeta(-n)*(n+1)*(2^(n+2) - 2), for n = 0 the limit is understood.

Original entry on oeis.org

1, 3, 2, -1, -2, 3, 6, -17, -34, 155, 310, -2073, -4146, 38227, 76454, -929569, -1859138, 28820619, 57641238, -1109652905, -2219305810, 51943281731, 103886563462, -2905151042481, -5810302084962, 191329672483963, 382659344967926, -14655626154768697
Offset: 0

Views

Author

Paul Curtz, Apr 06 2014

Keywords

Comments

Let G(m, n) denote the difference table of a(n):
1, 3, 2, -1, -2, 3, 6, -17, -34,...
2, -1, -3, -1, 5, 3, -23, -17,...
-3, -2, 2, 6, -2, -26, 6,...
1, 4, 4, -8, -24, 32,...
3, 0, -12, -16, 56,...
-3, -12, -4, 72,...
-9, 8, 76,...
17, 68,...
51,...
a(n) = G(0, n).
The main diagonal G(n, n) = 1, -1, 2, -8, 56, -608,... is essentially a signed version of A005439.
The first upper diagonal is the main diagonal multiplied by 3. G(n, n+1) = 3*G(n, n).
G(m, n) = G(m, n-1) + G(m+1,n-1).
Inverse binomial transform: after 1, 2, -3, A110501(n+1) is interleaved with 3*A110501(n+1), signed two by two. I. e. b(n) = 1, 2, -3, 1, 3, -3, -9, 17, 51,... . a(n+2) + b(n+2) = -1, 0, 1, 0, -3, 0, 17,... = A226158(n+2).
This is particular to the Genocchi numbers. If the first upper diagonal is proportional to the main diagonal (1, -1, 2, -8,...), the sequence and the inverse binomial transform are simply connected to the Genocchi numbers.

Crossrefs

Cf. A001469, A005439, A110501, A226158, A230324.

Programs

  • Maple
    A240485 := proc(n) if n = 0 then 1 elif n = 1 then 3 else
m := 2*iquo(n-1, 2) + 2; -2^irem(n-1, 2)*m*euler(m-1, 0) fi end:
seq(A240485(n), n=0..27); # Peter Luschny, Apr 09 2014
  • Mathematica
    a[n_] := Which[n == 0, 1, n == 1, 3, True, m = 2*Quotient[n-1, 2]+2; -2^Mod[n-1, 2]*m*EulerE[m-1, 0]]; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Apr 09 2014, after Peter Luschny *)
  • Sage
    def A240485(n):
    if n < 3: return [1,3,2][n]
    m = 2*((n+1)//2)
    b = 2*(1-2^m)*bernoulli(m)
    if is_even(n): b = 2*b
    return (-1)^ceil((n^2+1)/2)*b
[A240485(n) for n in (0..24)]  # Peter Luschny, Apr 08 2014

Formula

a(2*n+1) = a(2*n+2)/2 for n > 0.
-a(2*n+2)/2 = A226158(2*n+2) = A001469(n+1) = (2*n+2)*E(2*n+1, 0) where E(n, x) are the Euler polynomials.
a(n) = -2*A226158(n) - A226158(n+1).
E.g.f.: (2*exp(x)*(3*x+exp(x)*(2*x+1)+1))/(exp(x)+1)^2. - Peter Luschny, Apr 10 2014

A277627 Square array read by antidiagonals downwards: T(n,k), n>=0, k>=0, in which column 0 is equal to A057427: 0, 1, 1, 1, ..., and for k > 0 column k lists two zeros followed by the partial sums of column k-1.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 1, 4, 1, 0, 0, 0, 0, 0, 0, 3, 5, 1, 0, 0, 0, 0, 0, 0, 0, 6, 6, 1, 0, 0, 0, 0, 0, 0, 0, 1, 10, 7, 1, 0, 0, 0, 0, 0, 0, 0, 0, 4, 15, 8, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 21, 9, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 20, 28, 10, 1
Offset: 0

Views

Author

Paul Curtz, Oct 24 2016

Keywords

Comments

In other words, for n > 0 the column k lists 2*k+1 zeros together with the partial sums of the positive terms of column k-1. - Omar E. Pol, Oct 25 2016
Comments from the author:
1) ZSPEC =
0, 0, 0, 0, 0, 0, 0, 0, ...
1, 0, 0, 0, 0, 0, 0, 0, ...
1, 0, 0, 0, 0, 0, 0, 0, ...
1, 1, 0, 0, 0, 0, 0, 0, ...
1, 2, 0, 0, 0, 0, 0, 0, ...
1, 3, 1, 0, 0, 0, 0, 0, ...
1, 4, 3, 0, 0, 0, 0, 0, ...
1, 5, 6, 1, 0, 0, 0, 0, ...
etc.
The columns are the autosequences of the first kind of the title (column 1: 0, 0, followed by A001477(n); column 2: 0, 0, 0, 0, followed by A000217(n), etc) .
The positive terms are the Pascal triangle written by diagonals (A011973).
First column: A060576(n+1). Or A057427(n), n>-1, thanks to Omar E. Pol.
Row sums: A000045(n), autosequence of the first kind.
Alternated row sums and subtractions: 0, 1, 1, 0, -1, -1, 0 = A128834(n), autosequence of the first kind.
Antidiagonal sums: 0, 1, 1, 1, 2, 3, 4, 6, ... = A078012(n+2).
Application.
Numbers in triangle leading to the Genocchi numbers -A226158(n).
We multiply the columns of ZSPEC by d(n) = 1, -1, 2, -8, 56, -608, ... from A005439.
Hence, with only the first 0,
0,
1,
1,
1, -1,
1, -2,
1, -3, 2,
1, -4, 6,
1, -5, 12, -8,
1, -6, 20, -32,
1, -7, 30, -80, 56,
1, -8, 42, -160, 280,
etc.
The row sums is -A226158(n).
2) Now consider the case of the autosequences of the second kind.
First step.
2, 1, 1, 1, 1, 1, ... = A054977(n)
0, 0, 2, 3, 4, 5, 6, 7, ... = A199969(n) with offset 0
0, 0, 0, 0, 2, 5, 9, 14, 20, 27, ... see A000096
etc.
The positive terms are ASPEC in A191302. By triangle, they are either A029653(n) with A029653(0) = 2 instead of 1 or A029635(n).
Second step. YSPEC =
2, 0, 0, 0, 0, 0, ...
1, 0, 0, 0, 0, 0, ...
1, 2, 0, 0, 0, 0, ...
1, 3, 0, 0, 0, 0, ...
1, 4, 2, 0, 0, 0, ...
1, 5, 5, 0, 0, 0, ...
1, 6, 9, 2, 0, 0, ...
1, 7, 14, 7, 0, 0, ...
etc.
Diagonals by triangle: A029635(n).
This is the companion to ZSPEC.
Row sums: A000032(n), autosequence of the second kind.
Alternated row sums and subtractions: period 6 repeat 2, 1, -1, -2, -1, 1 = A087204(n), autosequence of the second kind.
Application.
Numbers in triangle leading to A230324(n), a companion to -A226158(n).
We multiply the columns of YSPEC by d(n) 1, -1, 2, -8, 56, ... (see above).
Hence, without zeros:
2,
1,
1, -2,
1, -3,
1, -4, 4,
1, -5, 10,
1, -6, 18, -16,
1, -7, 28, -56,
1, -8, 40, -128, 112,
1, -9, 54, -240, 504,
etc.
The row sum is A230324(n).

Links

Crossrefs

Cf. A011973 (without 0's), A007318 (Pascal's triangle).
Cf. A000045 (row sums), A078012 (antidiagonal sums).
Columns: A060576 or A057427 (k=0), A001477 (k=1), A000217 (k=2).
Cf. A000004, A000012, A000027, A000032, A005439, A029635, A029653, A054977, A087204, A128834, A191302, A199969, A226158, A230324.

Programs

  • Mathematica
    kMax = 13; col[0] = Join[{0}, Array[1&, kMax]]; col[k_] := col[k] = Join[{0, 0}, col[k-1][[1 ;; -3]] // Accumulate]; T[n_, k_] := col[k][[n+1]]; Table[T[n-k, k], {n, 0, kMax}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 15 2016 *)

Extensions

Better definition from Omar E. Pol, Oct 25 2016

A245683 Array T(n,k) read by antidiagonals, where T(0,k) = -A226158(k) and T(n+1,k) = 2*T(n,k+1) - T(n,k).

Original entry on oeis.org

0, 2, 1, 0, 1, 1, -6, -3, -1, 0, 0, -3, -3, -2, -1, 50, 25, 11, 4, 1, 0, 0, 25, 25, 18, 11, 6, 3, -854, -427, -201, -88, -35, -12, -3, 0, 0, -427, -427, -314, -201, -118, -65, -34, -17, 24930, 12465, 6019, 2796, 1241, 520, 201, 68, 17, 0
Offset: 0

Views

Author

Paul Curtz, Jul 29 2014

Keywords

Comments

Take T(n,k) = -A226158(k) and its transform via T(n+1,k) = 2*T(n,k+1) - T(n,k):
0, 1, 1, 0, -1, 0, 3, 0, -17, ...
2, 1, -1, -2, 1, 6, -3, -34, ... = A230324
0, -3, -3, 4, 11, -12, -65, ...
-6, -3, 11, 18, -35, -118, ...
0, 25, 25, -88, -201, ...
50, 25, -201, -314, ...
0, -427, -427, ...
-854, -427, ...
0, ...
Every row is alternatively an autosequence of the first kind, see A226158, and of the second kind, see A190339.
The second column is twice 1, -3, 25, -427, 12465, ... = (-1)^n*A009843(n) which is in the third column. See A132049(n), numerators of Euler's formula for Pi from the Bernoulli numbers, A243963 and A245244. Hence a link between the Genocchi numbers and Pi.
a(n) is the triangle of the increasing antidiagonals.

Examples

			Triangle a(n):
   0,
   2,  1,
   0,  1,  1,
  -6, -3, -1,  0,
   0, -3, -3, -2, -1,
  50, 25, 11,  4,  1,  0,
  etc.

Crossrefs

Cf. A226158, A230324, A009843, A132049, A243963, A245244.

Programs

  • Mathematica
    t[0, 0] = 0; t[0, 1] = 1; t[0, k_] := -k*EulerE[k-1, 0]; t[n_, k_] := t[n, k] = -t[n-1, k] + 2*t[n-1, k+1]; Table[t[n-k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 04 2014 *)
Showing 1-3 of 3 results.

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