A140218 -id:A140218 - cp's OEIS frontend

A140216 Numerators of the inverse triangle defined in A133135.

1, -1, 1, 1, -3, 1, -1, 3, -2, 1, 1, -5, 2, -5, 1, -1, 5, -1, 5, -3, 1, 1, -7, 1, 0, 3, -7, 1, -1, 7, -4, 0, 2, 7, -4, 1, 1, 3, 4, -21, -2, 21, 4, -9, 1, -1, -3, 13, 21, -26, -21, 10, 9, -5, 1, 1, -121, -13, 66, 26, -231, -10, 33, 5, -11, 1, -1, 121, -142, -66, 229, 231, -116, -33, 25, 11, -6

Offset: 0

Paul Curtz, Jun 22 2008

Row sums 1, 0, -1, 1, -6, 6, -8, 8, 2, -2,.. build in pairs.

			The triangle starts
1;
-1, 1;
1,-3, 1;
-1, 3,-2, 1;
1,-5, 2, -5, 1;
-1, 5,-1, 5, -3, 1;
1,-7, 1, 0, 3, -7, 1;
-1, 7,-4, 0, 2, 7,-4, 1;
1, 3, 4,-21, -2, 21, 4,-9, 1;
-1,-3,13, 21,-26,-21,10, 9,-5,1;
Pairs of numbers appear along adjacent diagonals. The first subdiagonal contains the sequence -A026741(n+2).

Cf. A133135, A140218.

Edited by R. J. Mathar, Aug 06 2008

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

Original entry on oeis.org

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

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A140216 Numerators of the inverse triangle defined in A133135.

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