A225825 -id:A225825 - cp's OEIS frontend

A227608 Denominators of A225825(n) difference table written by antidiagonals.

1, 2, 2, 6, 3, 6, 2, 3, 3, 2, 30, 15, 15, 15, 30, 2, 15, 15, 15, 15, 2, 42, 21, 105, 105, 105, 21, 42, 2, 21, 21, 105, 105, 21, 21, 2, 30, 15, 105, 105, 105, 105, 105, 15, 30, 2, 15, 15, 105, 105, 105, 105, 15, 15, 2, 66, 33, 165, 165, 1155, 231, 1155, 165, 165, 33, 66, 2, 33, 33, 165, 165, 231, 231, 165, 165, 33, 33, 2

Offset: 0

Paul Curtz, Aug 10 2013

			1,
-1/2,      1/2,
-1/6,     -2/3,     -1/6,
1/2,       1/3,     -1/3,     -1/2,
7/30,    11/15,    16/15,    11/15,     7/30,
-3/2,   -19/15,    -8/15,     8/15,    19/15,    3/2,
-31/42, -47/21, -368/105, -424/105, -368/105, -47/21, -31/42.
Row sums: 1, 0/2, -6/6, 0/6, 90/30, 0/30, -3570/210, 0/210, 32550/210,... .
Are the denominators A034386(n+1)?
Reduced row sums: 1, 0, -1, 0, 3, 0, -17, 0, 155,... = -A036968(n+1)? See A226158(n+2). First 100 terms checked by Jean-François Alcover.

Cf. A085738

max = 12; b[0] = 1; b[n_] := Numerator[ BernoulliB[n, 1/2] - (n+1)*EulerE[n, 0]]; t = Table[b[n], {n, 0, max}] / Table[ Sum[ Boole[ PrimeQ[d+1]]/(d+1), {d, Divisors[n]}] // Denominator, {n, 0, max}]; dt = Table[ Differences[t, n], {n, 0, max}]; Table[ dt[[n-k+1, k]] // Denominator, {n, 1, max}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 12 2013 *)

More terms from Jean-François Alcover, Aug 12 2013

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

Original entry on oeis.org

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

A230324 a(n) = A226158(n) - 2*A226158(n+1).

Original entry on oeis.org

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

Offset: 0

Paul Curtz, Oct 16 2013

sign

The array A(n,k) = A(n-1,k+1) - A(n-1,k) of the sequence in the first row and higher-order sequences in followup rows starts:
   2,  1,  -1,   -2,     1,     6,    -3, ...
  -1, -2,  -1,    3,     5,    -9,   -31, ...
  -1,  1,   4,    2,   -14,   -22,    82, ...
   2,  3,  -2,  -16,    -8,   104,   160, ...
   1, -5, -14,    8,   112,    56, -1160, ...
  -6, -9,  22,  104,   -56, -1216,  -608, ...
  -3, 31,  82, -160, -1160,   608, 18880, ...
etc.
a(n) is an autosequence: Its inverse binomial transform is the sequence (up to a sign), which means top row and left column in the difference array have the same absolute values.
The main diagonal is the double of the first upper diagonal: A(n,n) = 2*A(n,n+1).
A(n,n+1) = (-1)^n*A005439(n), which also appears as the first upper diagonal of the difference array of A226158(n).

			a(0) =  0 - 2 * (-1) =  2,
a(1) = -1 - 2 * (-1) =  1,
a(2) = -1 - 2 *   0  = -1,
a(3) =  0 - 2 *   1  = -2,
a(4) =  1 - 2 *   0  =  1,
a(5) =  0 - 2 * (-3) =  6.

G. C. Greubel, Table of n, a(n) for n = 0..500
OEIS Wiki, Autosequence

Cf. A050946.

A226158 := proc(n)
    if n = 0 then
        0;
    else
        Zeta(1-n)*2*n*(2^n-1) ;
    end if;
end proc:
A230324 := proc(n)
    A226158(n)-2*A226158(n+1) ;
end proc: # R. J. Mathar, Oct 28 2013

a[0] = 2; a[1] = 1; a[n_] := n EulerE[n-1, 0] - 2 (n+1) EulerE[n, 0];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 07 2017 *)

a(n)/2 + A164555(n)/A027642(n) = 2*A225825(n)/A141056(n).

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A227608 Denominators of A225825(n) difference table written by antidiagonals.

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

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A230324 a(n) = A226158(n) - 2*A226158(n+1).

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