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1-Euler triangle.
3

%I #12 Feb 03 2019 01:28:25

%S 1,1,1,1,1,1,1,5,5,1,1,14,30,14,1,1,33,146,146,33,1,1,72,603,1168,603,

%T 72,1,1,151,2241,7687,7687,2241,151,1,1,310,7780,44194,76870,44194,

%U 7780,310,1,1,629,25820,231236,649514,649514,231236,25820,629,1,1,1268,83121,1131504,4866222,7794168,4866222,1131504,83121,1268,1

%N 1-Euler triangle.

%C Formed with the same recurrence as the Euler triangle A008292 (adjusted for offset), but with the middle element of row n=2 set to 1.

%C Row sums are A188588. Second column is A188589.

%F T(n,k) = 0 if k < 0 or k > n.

%F T(n,k) = 1 if k=0 or k=n or n=2.

%F T(n,k)= (n-k+1)*T(n-1,k-1) + (k+1)*T(n-1,k), n > 2 and 1 <= k < n.

%e Triangle begins

%e 1;

%e 1, 1;

%e 1, 1, 1;

%e 1, 5, 5, 1;

%e 1, 14, 30, 14, 1;

%e 1, 33, 146, 146, 33, 1;

%e 1, 72, 603, 1168, 603, 72, 1;

%e 1, 151, 2241, 7687, 7687, 2241, 151, 1;

%e 1, 310, 7780, 44194, 76870, 44194, 7780, 310, 1;

%e 1, 629, 25820, 231236, 649514, 649514, 231236, 25820, 629, 1;

%p A188587 := proc(n,k) if k < 0 or k > n then 0; elif k=0 or k= n or n=2 then 1; else (n-k+1)*procname(n-1,k-1)+(k+1)*procname(n-1,k) ; end if; end proc:

%p seq(seq(A188587(n,k),k=0..n),n=0..10) ; # _R. J. Mathar_, Apr 13 2011

%K nonn,easy,tabl

%O 0,8

%A _Paul Barry_, Apr 04 2011