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%I #22 Apr 30 2014 01:31:34
%S 1,1,1,1,2,1,1,3,3,1,1,4,6,4,1,5,10,10,4,1,6,15,20,14,1,7,21,35,34,14,
%T 1,8,28,56,69,48,1,9,36,84,125,117,48,1,10,45,120,209,242,165,1,11,55,
%U 165,329,451,407,165,1,12,66,220,494,780,858,572,1,13,78,286,714,1274,1638,1430,572
%N Triangular array T read by rows: T(n,0) = 1 for n >= 0; T(n,k) = C(n,k) for k = 1,2,...,n, for n = 1,2,3; and for n >= 4, T(n,k) = T(n-1,k-1) + T(n-1,k) for k = 1,2,...,[ (n+2)/2 ] and T(n,(n+3)/2) = T(n-1,(n+1)/2) if n is odd.
%D E. Lucas, Théorie des Nombres, Albert Blanchard, Paris, 1958,tome1, p.88
%F T(n, k) = C(n, k) - C(n, k-4). - _Ralf Stephan_, Jan 09 2005
%F T(2n,n) = A026029(n). - _Philippe Deléham_, Mar 12 2013
%F T(2n-1,n) = A026016(n), n>0. - _Philippe Deléham_, Mar 12 2013
%e From _Philippe Deléham_, Mar 12 2013: (Start)
%e Triangle begins:
%e 1
%e 1, 1
%e 1, 2, 1
%e 1, 3, 3, 1
%e 1, 4, 6, 4
%e 1, 5, 10, 10, 4
%e 1, 6, 15, 20, 14
%e 1, 7, 21, 35, 34, 14
%e 1, 8, 28, 56, 69, 48
%e 1, 9, 36, 84, 125, 117, 48
%e 1, 10, 45, 120, 209, 242, 165
%e 1, 11, 55, 165, 329, 451, 407, 165
%e Pentagon arithmetic of Delannoy (in E. Lucas):
%e 1, 1, 1, 1, 0
%e 1, 2, 3, 4, 4, 0
%e 1, 3, 6, 10, 14, 14, 0
%e 1, 4, 10, 20, 34, 48, 48, 0
%e 1, 5, 15, 35, 69, 117, 165, 165,
%e 1, 6, 21, 56, 125, 242, 407, 572,
%e 1, 7, 28, 84, 209, 451, 858, 1430 (End)
%o (PARI) {T(n, k) = if( 2*k < n+4, binomial( n, k) - binomial( n, k-4), 0)} /* _Michael Somos_, Jan 08 2012 */
%K nonn,tabf
%O 1,5
%A _Clark Kimberling_
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