A239322 - OEIS

%I #36 Feb 06 2023 14:21:46

%S 0,1,1,-5,-11,61,211,-1385,-6551,50521,303271,-2702765,-19665491,

%T 199360981,1704396331,-19391512145,-190473830831,2404879675441,

%U 26684005437391,-370371188237525

%N Interleave (-1)^n*(A000182(n+1) - A000364(n)), -A028296(n+1).

%C Difference table of a(n):

%C 0,         1,     1,    -5,  -11,    61,   211, -1385,...

%C 1,         0,    -6,    -6,   72,   150, -1596,...

%C -1,       -6,     0,    78,   78, -1746,...

%C -5,       -6,    78,     0, -1824,...

%C 11,       72,    78, -1824,...

%C 61,     -150, -1746,...

%C -211,  -1596,...

%C -1385,...

%C etc.

%C a(n) is an autosequence (its inverse binomial transform is the signed sequence) of the first kind (its main diagonal is A000004=0's and the first two upper diagonal are the same). Like A000045(n).

%C Note that e(n) = A000111(n+1) - A000111(n) = 0, 0, 1, 3, 11, 45, 211,... is not in the OEIS. a(2n) = (-1)*(n+1)*e(2n).

%C Antidiagonals upon A000004:

%C 1,

%C 1,

%C -6,  -5,

%C -6, -11,

%C 78,  72,  61,

%C 78, 150, 211,

%C Row sum: 1, 1, -11, -17, 211, 439,... .

%C b(n) = a(n) mod 9 = 0 followed by period 6: repeat 1, 1, 4, 7, 7, 4 is also an autosequence of the first kind.

%e a(0)=1-1=0, a(1)=-(-1)=1, a(2)=2-1=1, a(3)=-5, a(4)=-(16-5)=-11.

%Y Cf. Zig (A000364) and Zag (A000182) give Euler A000111(n).

%K sign

%O 0,4

%A _Paul Curtz_, Mar 28 2014