A198631 -id:A198631 - cp's OEIS frontend

A243868 0 followed by -(n+1)*A226158(n).

0, 0, 2, 3, 0, -5, 0, 21, 0, -153, 0, 1705, 0, -26949, 0, 573405, 0, -15802673, 0, 547591761, 0, -23302711005, 0, 1194695479813, 0, -72628776062025, 0, 5165901157067001, 0, -425013158488292213, 0

Offset: 0

Paul Curtz, Jun 13 2014

An autosequence is a sequence which has its inverse binomial transform equal to the signed sequence. If the main diagonal is A000004=0's it is of the first kind. It is of the second kind if the main diagonal is the upper diagonal multiplied by 2.
Starting from the autosequence of second kind A198631(n)/A006519(n+1),the fractional Euler numbers,we build a family of alternated sequences of second and first kind. A row is 0 followed by n+1 times the preceding one.
1, 1/2, 0, -1/4,  0, 1/2, 0, -17/8,   0, 31/2,...
0,   1, 1,    0, -1,   0, 3,     0, -17,    0, 155,... = -A226158(n)
0,   0, 2,    3,  0,  -5, 0,    21,   0, -153, 0, 1705,... = a(n).
a(n) is an autosequence of the second kind. Its difference table is:
0,     0,   2,   3,   0,  -5,   0,  21, 0, -153,...
0,     2,   1,  -3,  -5,   5,  21, -21,...
2,    -1,  -4,  -2,  10,  16, -42,...
-3,   -3,   2,  12,   6, -58,..
0,     5,  10,  -6, -64,...
5,     5, -16, -58,...
0,   -21, -42,...
-21, -21,...
0,... .
a(n) is a post Genocchi sequence.

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

Cf. A102055, A230011, A229054.

a[0] = a[1] = 0; a[2] = 2; a[n_] := -n*(n-1)*EulerE[n-2, 0]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 17 2014 *)

The fourth column of the second triangle of A133135(n) (see also A140218) is
1, -5/2, 5/2, 0, 0, -21/2, 21/2,... = b(n).
c(n) = 0, 0, 0, 0, followed by 2*(-1)^n*b(n) = 0, 0, 0, 0, 2, 5, 5, 0, 0, 21, 21, -132, -132,... . Autosequence.
a(n) = c(n+2) -c(n+1).

More terms from Jean-François Alcover, Jun 17 2014

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

cp's OEIS Frontend

A243868 0 followed by -(n+1)*A226158(n).

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