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Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(0,0) = 1; T(0,k) = 0 if k < 0 or if k > 0; T(n,k) = k*T(n-1,k-1) + (k+2)*T(n-1,k+1).
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%I #12 Aug 18 2017 03:15:39

%S 1,0,1,2,0,2,0,8,0,6,16,0,40,0,24,0,136,0,240,0,120,272,0,1232,0,1680,

%T 0,720,0,3968,0,12096,0,13440,0,5040,7936,0,56320,0,129024,0,120960,0,

%U 40320,0,176896,0,814080,0,1491840,0,1209600,0,362880,353792,0

%N Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(0,0) = 1; T(0,k) = 0 if k < 0 or if k > 0; T(n,k) = k*T(n-1,k-1) + (k+2)*T(n-1,k+1).

%C Triangle is related to the tangent numbers A000182.

%D S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 446.

%F T(n, n) = n!; T(n, 0) = 0 if n = 2m+1; T(n, 0) = A000182(m+1) if n = 2m.

%F Sum_{k>=0} T(m, k)*T(n, k)*(k+1) = T(m+n, 0).

%F Sum_{k>=0} T(n, k) = |A003707(n+1)|.

%e Triangle begins:

%e 1;

%e 0, 1;

%e 2, 0, 2;

%e 0, 8, 0, 6;

%e 16, 0, 40, 0, 24;

%e 0, 136, 0, 240, 0, 120;

%e 272, 0, 1232, 0, 1680, 0, 720;

%e 0, 3968, 0, 12096, 0, 13440, 0, 5040;

%e 7936, 0, 56320, 0, 129024, 0, 120960, 0, 40320;

%e 0, 176896, 0, 814080, 0, 1491840, 0, 1209600, 0, 362880;

%e 353792, 0, 3610112, 0, 12207360, 0, 18627840, 0, 13305660, 0, 3628800;

%e ...

%p T:=proc(n,k) if k=-1 then 0 elif n=1 and k=1 then 1 elif k>n then 0 else (k-1)*T(n-1,k-1)+(k+1)*T(n-1,k+1) fi end: for n from 1 to 11 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form [Produces triangle with a different offset] # _Emeric Deutsch_, Jun 13 2005

%Y Similar to A104035. Leading edge is essentially A000182.

%Y Cf. A003707.

%K nonn,easy,tabl

%O 0,4

%A _N. J. A. Sloane_, Jun 10 2005

%E More terms from _Emeric Deutsch_, Jun 13 2005

%E Additional comments from _Philippe Deléham_, Sep 17 2005

%E Edited by _N. J. A. Sloane_, Aug 23 2008 at the suggestion of _R. J. Mathar_