A235774 -id:A235774 - cp's OEIS frontend

A239315 Array read by antidiagonals: denominators of the core of the classical Bernoulli numbers.

15, 15, 15, 105, 105, 105, 21, 105, 105, 21, 105, 105, 105, 105, 105, 15, 105, 105, 105, 105, 15, 165, 165, 1155, 231, 1155, 165, 165, 33, 165, 165, 231, 231, 165, 165, 33, 15015, 15015, 15015, 15015, 15015, 15015, 15015, 15015, 15015

Offset: 0

Paul Curtz, Mar 15 2014

We consider the autosequence A164555(n)/A027642(n) (see A190339(n)) and its difference table without the first two rows and the first two columns:
2/15,     1/15,    -1/105,   -1/21,   -1/105,     1/15,   7/165,   -5/33,...
-1/15,  -8/105,    -4/105,   4/105,    8/105,   -4/165, -32/165,...
-1/105,  4/105,     8/105,   4/105, -116/1155, -28/165,...
1/21,    4/105,    -4/105, -32/231,   -16/231,...
-1/105, -8/105, -116/1155,  16/231,...
-1/15,  -4/165,    28/165,...
7/165,  32/165,...
5/33,... etc.
This is an autosequence of the second kind.
The antidiagonals are palindromes in absolute values.
a(n) are the denominators. Multiples of 3.
Sum of odd antidiagonals: 2/15, -2/21, 2/15, -10/33, 1382/1365,... = -2*A000367(n+2)/A001897(n+2).
The sum of the even antidiagonals is A000004.
2/15, 0, -2/21,... = -4*A027641(n+4)/A027642(n+4) = -4*A164555(n)/A027642(n+4) and others.

			As a triangle:
15,
15,   15,
105, 105, 105,
21,  105, 105, 21,
105, 105, 105, 105, 105,
etc.

Cf. A085737/A085738, A168516/A168426 (autosequence), A027641, A176327/A176289, A235774, A165161/A051717(n+1).

max = 12; tb = Table[BernoulliB[n], {n, 0, max}]; td = Table[Differences[tb, n][[3 ;; -1]], {n, 2, max - 1}]; Table[td[[n - k + 1, k]] // Denominator, {n, 1, max - 3}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 11 2014 *)

A238398 Numerators of inverse binomial transform of A198631(n)/A006519(n+1) with -1 instead of A198631(1)=1.

Original entry on oeis.org

1, -3, 2, -11, 4, -11, 6, -39, 8, -49, 10, 647, 12, -5487, 14, 929329, 16, -3202325, 18, 221930505, 20, -4722116563, 22, 968383680643, 24, -14717667114201, 26, 2093660879252563, 28, -86125672563201239, 30, 129848163681107300961, 32

Offset: 0

Paul Curtz, Feb 26 2014

sign

From modified fractional Euler numbers.
Inverse binomial transform:
1, -3/2, 2, -11/4, 4, -11/2, 6, -39/8, 8, -49/2, 10, 647/4, 12, -5487/2,... = a(n)/b(n).  b(2n) = A004277(n).
Difference table of c(n) = 1, -1/2, 0, -1/4,... :
1,    -1/2,    0,  -1/4,     0,    1/2,      0,...
-3/2,  1/2, -1/4,   1/4,   1/2,   -1/2,  -17/8,...
2,    -3/4,  1/2,   1/4,    -1,  -13/8,   17/4,...
-11/4, 5/4, -1/4,  -5/4,  -5/8,   47/8,   73/8,...
4,    -3/2,   -1,   5/8,  13/2,   13/4, -107/2,...
-11/2, 1/2, 13/8,  47/8, -13/4, -227/4, -227/8,
6,     9/8, 17/4, -73/8, -107/2, 227/8, 2957/4,...
etc.
c(n) + a(n)/b(n) = 2, -2, 2, -3, 4, -5, 6, -7, 8, -9,... = A233583(n+1) signed. (a(n) discovered in 2013)

Cf. A235774.

max = 40;(* b = A198631 *) b[0] = 1; b[1] = -1; b[n_] := Numerator[EulerE[n, 1]/(2^n-1)]; bb = Table[b[n]/2^IntegerExponent[n+1, 2], {n, 0, max}]; a[n_] := Differences[bb, n] // First // Numerator ; Table[a[n], {n, 0, max}]

A244237 Numerators of the inverse binomial transform of (-1 followed by A164555(n+1)/A027642(n+1)).

Original entry on oeis.org

-1, 3, -11, 2, -61, 2, -83, 2, -61, 2, -127, 2, -6151, 2, -5, 2, -4637, 2, 42271, 2, -175241, 2, 854237, 2, -236369551, 2, 8553091, 2, -23749462769, 2, 8615841247361, 2, -7709321042237, 2, 2577687858355, 2, -26315271553057315753, 2

Offset: 0

Paul Curtz, Jun 23 2014

See A244213. (The binomial transform of A198631(n)/A006519(n+1) is A143074(n)/A006519(n+1)).
Difference table of -1 followed by A164555(n+1)/A027642(n+1), see A190339:
      -1,   1/2,    1/6,      0,  -1/30,     0,  1/42, 0,...
     3/2,  -1/3,   -1/6,  -1/30,   1/30,  1/42, -1/42,...
   -11/6,   1/6,   2/15,   1/15, -1/105, -1/21,...
       2, -1/30,  -1/15, -8/105, -4/105,...
  -61/30, -1/30, -1/105,  4/105,...
       2,  1/42,   1/21,...
  -83/42,  1/42,...
       2,...
  etc.
The corresponding denominators to a(n) are A027642(n). See A085738.
From the second Bernoulli numbers.

Cf. A190339, A235774, A244213, A085738, A164555, A027642, A143074, A006519.

(A164555(n+2) - a(n+2))/A027642(n+2) = (-1)^n*2.

a(12)-a(37) from Jean-François Alcover

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A239315 Array read by antidiagonals: denominators of the core of the classical Bernoulli numbers.

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A238398 Numerators of inverse binomial transform of A198631(n)/A006519(n+1) with -1 instead of A198631(1)=1.

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A244237 Numerators of the inverse binomial transform of (-1 followed by A164555(n+1)/A027642(n+1)).

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