cp's OEIS Frontend

A107729 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).

%I A107729 #12 Aug 18 2017 03:15:39
%S A107729 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 A107729 0,720,0,3968,0,12096,0,13440,0,5040,7936,0,56320,0,129024,0,120960,0,
%U A107729 40320,0,176896,0,814080,0,1491840,0,1209600,0,362880,353792,0
%N A107729 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 A107729 Triangle is related to the tangent numbers A000182.
%D A107729 S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 446.
%F A107729 T(n, n) = n!; T(n, 0) = 0 if n = 2m+1; T(n, 0) = A000182(m+1) if n = 2m.
%F A107729 Sum_{k>=0} T(m, k)*T(n, k)*(k+1) = T(m+n, 0).
%F A107729 Sum_{k>=0} T(n, k) = |A003707(n+1)|.
%e A107729 Triangle begins:
%e A107729      1;
%e A107729      0,    1;
%e A107729      2,    0,    2;
%e A107729      0,    8,    0,    6;
%e A107729     16,    0,   40,    0,    24;
%e A107729      0,   136,   0,   240,    0,   120;
%e A107729     272,   0,  1232,   0,   1680,   0,    720;
%e A107729      0,  3968,   0,  12096,   0,  13440,   0,  5040;
%e A107729    7936,   0,  56320,  0,  129024,  0,  120960,  0,   40320;
%e A107729      0, 176896,  0, 814080,   0, 1491840,  0, 1209600,  0, 362880;
%e A107729   353792,  0, 3610112, 0, 12207360, 0, 18627840, 0, 13305660, 0, 3628800;
%e A107729   ...
%p A107729 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 A107729 Similar to A104035. Leading edge is essentially A000182.
%Y A107729 Cf. A003707.
%K A107729 nonn,easy,tabl
%O A107729 0,4
%A A107729 _N. J. A. Sloane_, Jun 10 2005
%E A107729 More terms from _Emeric Deutsch_, Jun 13 2005
%E A107729 Additional comments from _Philippe Deléham_, Sep 17 2005
%E A107729 Edited by _N. J. A. Sloane_, Aug 23 2008 at the suggestion of _R. J. Mathar_