0,13

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

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

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

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

Row sums give A016116(n+2) - Yosu Yurramendi (yosu.yurramendi(AT)ehu.es), Aug 07 2008

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

Contribution from Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Dec 04 2009: (Start)

The sequence of coefficients of a general polynomial recursion that links at w=2 to the Pascal triangle is here w=0.

Row sums are:

{1, 2, 2, 4, 4, 8, 8, 16, 16, 32, 32, 64,...} (End)

S. J. Cyvin et al., Unbranched catacondensed polygonal systems containing hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774.

M. E. Horn, The Didactical Relevance of the Pauli Pascal Triangle [From Michael Somos]

T(n, k)=T(n-1, k-1)+T(n-1, k) if n odd or k even, else 0. T(0, 0)=1.

T(n, k)=T(n-2, k-2)+T(n-2, k). T(0, 0)=T(1, 0)=T(1, 1)=1.

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

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

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]

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]

Contribution from Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Dec 04 2009: (Start)

{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, 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, 5, 10, 10, 10, 10, 5, 5, 1, 1} (End)

Contribution from Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Dec 04 2009: (Start)

Clear[p, n, x, a]

w = 0;

p[x, 1] := 1;

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]]

a = Table[CoefficientList[p[x, n], x], {n, 1, 12}]

Flatten[a] (End)

(PARI) {T(n, k) = binomial(n%2, k%2) * binomial(n\2, k\2)} [From Michael Somos]

Cf. A007318. A051160(n, k)=(-1)^[ k/2 ]*A051159(n, k).

Cf. A016116, A034851.

Sequence in context: A035196 A158020 A051160 this_sequence A035697 A135549 A124737

Adjacent sequences: A051156 A051157 A051158 this_sequence A051160 A051161 A051162

Cf. A169623 [From Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Dec 04 2009]

nonn,tabl,easy,nice,new

Michael Somos