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%I #16 Jun 21 2014 16:15:48
%S 0,0,2,3,0,-5,0,21,0,-153,0,1705,0,-26949,0,573405,0,-15802673,0,
%T 547591761,0,-23302711005,0,1194695479813,0,-72628776062025,0,
%U 5165901157067001,0,-425013158488292213,0
%N 0 followed by -(n+1)*A226158(n).
%C 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.
%C 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.
%C 1, 1/2, 0, -1/4, 0, 1/2, 0, -17/8, 0, 31/2,...
%C 0, 1, 1, 0, -1, 0, 3, 0, -17, 0, 155,... = -A226158(n)
%C 0, 0, 2, 3, 0, -5, 0, 21, 0, -153, 0, 1705,... = a(n).
%C a(n) is an autosequence of the second kind. Its difference table is:
%C 0, 0, 2, 3, 0, -5, 0, 21, 0, -153,...
%C 0, 2, 1, -3, -5, 5, 21, -21,...
%C 2, -1, -4, -2, 10, 16, -42,...
%C -3, -3, 2, 12, 6, -58,..
%C 0, 5, 10, -6, -64,...
%C 5, 5, -16, -58,...
%C 0, -21, -42,...
%C -21, -21,...
%C 0,... .
%C a(n) is a post Genocchi sequence.
%F The fourth column of the second triangle of A133135(n) (see also A140218) is
%F 1, -5/2, 5/2, 0, 0, -21/2, 21/2,... = b(n).
%F 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.
%F a(n) = c(n+2) -c(n+1).
%e 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.
%t 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 *)
%Y Cf. A102055, A230011, A229054.
%K sign,frac,tabl
%O 0,3
%A _Paul Curtz_, Jun 13 2014
%E More terms from _Jean-François Alcover_, Jun 17 2014
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