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A051159 Triangular array made of three copies of Pascal's triangle. 21
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, 1, 0, 6, 0, 15, 0, 20, 0, 15, 0, 6, 0, 1, 1, 1, 6, 6, 15, 15, 20, 20, 15, 15, 6, 6, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,13

COMMENTS

Computing each term modulo 2 also gives A047999, i.e., a(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, 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 Deléham, Aug 02 2006

Number of n-bead black-white reversible strings with k black beads; also binary grids; string is palindromic. - Yosu Yurramendi, Aug 07 2008

Row sums give A016116(n+1). - Yosu Yurramendi, Aug 07 2008 [corrected by Petros Hadjicostas, Nov 04 2017]

Coefficients in expansion of (x + y)^n where x and y anticommute (y x = -x y), that is, q-binomial coefficients when q = -1. - Michael Somos, Feb 16 2009

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, ...}. - Roger L. Bagula and Gary W. Adamson, Dec 04 2009

T(n,k) is the number of palindromic compositions of n+1 with exactly k+1 parts. T(6,4) = 3 because we have the following compositions of n+1=7 with length k+1=5: 1+1+3+1+1, 2+1+1+1+2, 1+2+1+2+1. - Geoffrey Critzer, Mar 15 2014 [corrected by Petros Hadjicostas, Nov 03 2017]

Let P(n,k) be the number of palindromic compositions of n with exactly k parts. MacMahon (1893) was the first to prove that P(n,k) = T(n-1,k-1), where T(n,k) are the numbers in this sequence (see the comment above by G. Critzer). He actually proved that, for 1 <= s <= m, we have P(2*m,2*s) = P(2*m,2*s-1) = P(2*m-1, 2*s-1) = bin(m-1, s-1), but  P(2*m-1, 2*s) = 0. For the current sequence, this can be translated into T(2*m-1, 2*s-1) = T(2*m-1,2*s-2) = T(2*m-2, 2*s-2) = bin(m-1,s-1), but T(2m-2, 2*s-1) = 0 (valid again for 1 <= s <= m). - Petros Hadjicostas, Nov 03 2017

T is the infinite lower triangular matrix for this sequence; define two others, U and V; let U(n,k)=e_k(-1,2,-3,...,(-1)^n n), where e_k is the k-th elementary symmetric polynomial, and let V be the diagonal matrix A057077 (periodic sequence 1,1,-1,-1). Clearly V^-1 = V. Conjecture: U = U^-1, T = U . V, T^-1 = V . U, and |T| = |U|. - George Beck, Dec 16 2017

Let T*(n,k)=T(n,k) except when n is odd and k=(n+1)/2, where T*(n,k) = T(n,k)+2^((n-1)/2). Thus, T*(n,k) is the number of non-isomorphic symmetric stairs with n cells and k steps, i.e., k-1 changes of direction. See A016116. - Christian Barrientos and Sarah Minion, Jul 29 2018

LINKS

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.

E. Burlachenko, Fractal generalized Pascal matrices, arXiv:1612.00970 [math.NT], 2016. See p. 3.

S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Unbranched catacondensed polygonal systems containing hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774. See Table 1 on p. 763.

M. E. Horn, The Didactical Relevance of the Pauli Pascal Triangle, arXiv:physics/0611277 [physics.ed-ph], 2006. [Michael Somos]

F. Al-Kharousi, R. Kehinde, A. Umar, Combinatorial results for certain semigroups of partial isometries of a finite chain, The Australasian Journal of Combinatorics, Volume 58 (3) (2014), 363-375.

P. A. MacMahon, Memoir on the Theory of the Compositions of Numbers, Phil. Trans. Royal Soc. London A, 184 (1893), 835-901.

Index entries for triangles and arrays related to Pascal's triangle

FORMULA

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, Aug 21 2004

Sum_{k=0..n} k * T(n,k) = A093968(n); A093968 = S-D transform of A001477. - Philippe Deléham, Aug 02 2006

Equals 2*A034851 - A007318. - Gary W. Adamson, Dec 31 2007. [Corrected by Yosu Yurramendi, Aug 07 2008]

A051160(n, k) = (-1)^floor(k/2) * T(n, k).

Sum_{k = 0..n} T(n,k)*x^k = A000012(n), A016116(n+1), A056487(n), A136859(n+2) for x = 0, 1, 2, 3 respectively. - Philippe Deléham, Mar 11 2014

G.f.: (1+x+x*y)/(1-x^2-y^2*x^2). - Philippe Deléham, Mar 11 2014

EXAMPLE

Triangle starts:

{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}

... - Roger L. Bagula and Gary W. Adamson, Dec 04 2009

MAPLE

T:= proc(n, k) option remember; `if`(n=0 and k=0, 1,

      `if`(n<0 or k<0, 0, `if`(irem(n, 2)=1 or

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

    end:

seq(seq(T(n, k), k=0..n), n=0..14);  # Alois P. Heinz, Jul 12 2014

MATHEMATICA

T[ n_, k_] := QBinomial[n, k, -1]; (* Michael Somos, Jun 14 2011; since V7 *)

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] (* Roger L. Bagula and Gary W. Adamson, Dec 04 2009 *)

PROG

(PARI) {T(n, k) = binomial(n%2, k%2) * binomial(n\2, k\2)};

(Haskell)

a051159 n k = a051159_tabl !! n !! k

a051159_row n = a051159_tabl !! n

a051159_tabl = [1] : f [1] [1, 1] where

   f us vs = vs : f vs (zipWith (+) ([0, 0] ++ us) (us ++ [0, 0]))

-- Reinhard Zumkeller, Apr 25 2013

CROSSREFS

Cf. A007318. A051160.

Cf. A016116, A034851, A169623.

Cf. A036355.

Sequence in context: A287475 A158020 A051160 * A035697 A135549 A262666

Adjacent sequences:  A051156 A051157 A051158 * A051160 A051161 A051162

KEYWORD

nonn,tabl,easy,nice

AUTHOR

Michael Somos, Oct 14 1999

STATUS

approved

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Last modified November 13 22:50 EST 2018. Contains 317149 sequences. (Running on oeis4.)