%I A051159
%S A051159 1,1,1,1,0,1,1,1,1,1,1,0,2,0,1,1,1,2,2,1,1,1,0,3,0,3,0,1,1,1,3,3,3,3,1,
%T A051159 1,1,0,4,0,6,0,4,0,1,1,1,4,4,6,6,4,4,1,1,1,0,5,0,10,0,10,0,5,0,1,1,1,5,
%U A051159 5,10,10,10,10,5,5,1,1,1,0,6,0,15,0,20,0,15,0,6,0,1,1,1,6,6,15,15
%N A051159 Triangular array made of three copies of Pascal's triangle.
%C A051159 Computing each term modulo 2 also gives A047999, i.e. A051159[n] mod 
               2 = A007318[n] mod 2 for all n. (The triangle is paritywise isomorphic 
               to Pascal's Triangle) - Antti Karttunen
%C A051159 5th row/column gives entries of A000217 (triangular numbers C(n+1,2)) 
               repeated twice and every other entry in 6th row/column form A000217. 
               7th row/column gives entries of A000292 (Tetrahedral (or pyramidal) 
               nos: C(n+3,3)) repeated twice and every other entry in 8th row/column 
               form A000292. 9th row/column gives entries of A000332 (binomial coefficients 
               binomial(n,4)) repeated twice and every other entry in 10th row/column 
               form A000332. 11th row/column gives entries of A000389 (binomial 
               coefficients C(n,5)) repeated twice and every other entry in 12th 
               row/column form A000389. - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), 
               Aug 21 2004
%C A051159 If Sum_{k=0..n}A(k)*T(n,k)=B(n), the sequence B is the S-D transform 
               of the sequence A . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Aug 02 2006
%C A051159 Number of n-bead black-white reversible strings with k black beads; also 
               binary grids; string is palindromic. - Yosu Yurramendi (yosu.yurramendi(AT)ehu.es), 
               Aug 07 2008
%C A051159 Row sums give A016116(n+2) - Yosu Yurramendi (yosu.yurramendi(AT)ehu.es), 
               Aug 07 2008
%C A051159 Coefficients of expansion of (x+y)^n where x and y anticommute (yx = 
               -xy), that is, q-binomial coefficients when q = -1. - Michael Somos 
               Feb 16 2009
%C A051159 Contribution from Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), 
               Dec 04 2009: (Start)
%C A051159 The sequence of coefficients of a general polynomial recursion that links 
               at w=2 to the Pascal triangle is here w=0.
%C A051159 Row sums are:
%C A051159 {1, 2, 2, 4, 4, 8, 8, 16, 16, 32, 32, 64,...} (End)
%D A051159 S. J. Cyvin et al., Unbranched catacondensed polygonal systems containing 
               hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774.
%H A051159 M. E. Horn, <a href="http://arXiv.org/abs/physics/0611277">The Didactical 
               Relevance of the Pauli Pascal Triangle</a> [From Michael Somos]
%F A051159 T(n, k)=T(n-1, k-1)+T(n-1, k) if n odd or k even, else 0. T(0, 0)=1.
%F A051159 T(n, k)=T(n-2, k-2)+T(n-2, k). T(0, 0)=T(1, 0)=T(1, 1)=1.
%F A051159 Square array made by setting first row/column to 1's (A(i, 0) = A(0, 
               j) = 1); A(1, 1) = 0; A(1, j) = A(1, j-2); A(i, 1) = A(i-2, 1); other 
               entries A(i, j) = A(i-2, j) + A(i, j-2). - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), 
               Aug 21 2004
%F A051159 Sum_{k=0..n}k*T(n,k)=A093968(n); A093968 = S-D transform of A001477 . 
               - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 02 2006
%F A051159 Equals 2*A034851 - A007318, - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Dec 31 2007. [Corrected by Yosu Yurramendi (yosu.yurramendi(AT)ehu.es), 
               Aug 07 2008]
%F A051159 w-0:\q p(x,n)=If[Mod[n, 2] == 0, (x + 1)*p(x, n - 1), (x^2 + w*x + 1)^Floor[n/
               2]] [From Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), 
               Dec 04 2009]
%e A051159 Contribution from Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), 
               Dec 04 2009: (Start)
%e A051159 {1},
%e A051159 {1, 1},
%e A051159 {1, 0, 1},
%e A051159 {1, 1, 1, 1},
%e A051159 {1, 0, 2, 0, 1},
%e A051159 {1, 1, 2, 2, 1, 1},
%e A051159 {1, 0, 3, 0, 3, 0, 1},
%e A051159 {1, 1, 3, 3, 3, 3, 1, 1},
%e A051159 {1, 0, 4, 0, 6, 0, 4, 0, 1},
%e A051159 {1, 1, 4, 4, 6, 6, 4, 4, 1, 1},
%e A051159 {1, 0, 5, 0, 10, 0, 10, 0, 5, 0, 1},
%e A051159 {1, 1, 5, 5, 10, 10, 10, 10, 5, 5, 1, 1} (End)
%t A051159 Contribution from Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), 
               Dec 04 2009: (Start)
%t A051159 Clear[p, n, x, a]
%t A051159 w = 0;
%t A051159 p[x, 1] := 1;
%t A051159 p[x_, n_] := p[x, n] = If[Mod[n, 2] == 0, (x + 1)*p[x, n - 1], (x^2 + 
               w*x + 1)^Floor[n/2]]
%t A051159 a = Table[CoefficientList[p[x, n], x], {n, 1, 12}]
%t A051159 Flatten[a] (End)
%o A051159 (PARI) {T(n, k) = binomial(n%2, k%2) * binomial(n\2, k\2)} [From Michael 
               Somos]
%Y A051159 Cf. A007318. A051160(n, k)=(-1)^[ k/2 ]*A051159(n, k).
%Y A051159 Cf. A016116, A034851.
%Y A051159 Sequence in context: A035196 A158020 A051160 this_sequence A035697 A135549 
               A124737
%Y A051159 Adjacent sequences: A051156 A051157 A051158 this_sequence A051160 A051161 
               A051162
%Y A051159 Cf. A169623 [From Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), 
               Dec 04 2009]
%K A051159 nonn,tabl,easy,nice,new
%O A051159 0,13
%A A051159 Michael Somos