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A113413 A Riordan array of coordination sequences. 10
1, 2, 1, 2, 4, 1, 2, 8, 6, 1, 2, 12, 18, 8, 1, 2, 16, 38, 32, 10, 1, 2, 20, 66, 88, 50, 12, 1, 2, 24, 102, 192, 170, 72, 14, 1, 2, 28, 146, 360, 450, 292, 98, 16, 1, 2, 32, 198, 608, 1002, 912, 462, 128, 18, 1, 2, 36, 258, 952, 1970, 2364, 1666, 688, 162, 20, 1, 2, 40, 326 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Columns include A040000, A008574, A005899, A008412, A008413, A008414. Row sums are A078057(n)=A001333(n+1). Diagonal sums are A001590(n+3). Reverse of A035607. Signed version is A080246. Inverse is A080245.

For another version see A122542. - Philippe Deléham, Oct 15 2006

T(n,k) is the number of length n words on alphabet {0,1,2} with no two consecutive 1's and no two consecutive 2's and having exactly k 0's. - Geoffrey Critzer, Jun 11 2015

From Eric W. Weisstein, Feb 17 2016: (Start)

Triangle of coefficients (from low to high degree) of x^-n * vertex cover polynomial of the n-ladder graph P_2 \square p_n:

  Psi_{L_1}: x*(2 + x) -> {2, 1}

  Psi_{L_2}: x^2*(2 + 4 x + x^2) -> {2, 4, 1}

  Psi_{L_3}: x^3*(2 + 8 x + 6 x^2 + x^3) -> {2, 8, 6, 1}

(End)

LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows

Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Sect. 2.3.

Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.

Mirka Miller, Hebert Perez-Roses, and Joe Ryan, The Maximum Degree-and-Diameter-Bounded Subgraph in the Mesh, arXiv preprint arXiv:1203.4069 [math.CO], 2012.

FORMULA

From Paul Barry, Nov 13 2005: (Start)

Riordan array ((1+x)/(1-x), x(1+x)/(1-x)).

T(n, k) = sum{i=0..n-k, C(k+1, i)C(n-i, k)}.

T(n, k) = sum{j=0..n-k, C(k+j, j)C(k+1, n-k-j)}.

T(n, k) = D(n, k) + D(n-1, k) where D(n, k) = sum{j=0..n-k, C(n-k, j) C(k, j)2^j} = A008288(n, k).

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

T(n, k) = sum{j=0..n, C(floor((n+j)/2), k)C(k, floor((n-j)/2))}. (End)

T(n, k) = C(n, k)*hypergeometric([-k-1, k-n], [-n], -1). - Peter Luschny, Sep 17 2014

EXAMPLE

Triangle begins

1;

2, 1;

2, 4, 1;

2, 8, 6, 1;

2, 12, 18, 8, 1;

2, 16, 38, 32, 10, 1;

2, 20, 66, 88, 50, 12, 1;

MATHEMATICA

nn = 10; Map[Select[#, # > 0 &] &, CoefficientList[Series[1/(1 - 2 x/(1 + x) - y x), {x, 0, nn}], {x, y}]] // Grid (* Geoffrey Critzer, Jun 11 2015 *)

CoefficientList[CoefficientList[Series[1/(1 - 2 x/(1 + x) - y x), {x, 0, 10}, {y, 0, 10}], x], y] (* Eric W. Weisstein, Feb 17 2016 *)

PROG

(Sage)

T = lambda n, k : binomial(n, k)*hypergeometric([-k-1, k-n], [-n], -1).simplify_hypergeometric()

A113413 = lambda n, k : 1 if n==0 and k==0 else T(n, k)

for n in (0..12): print [A113413(n, k) for k in (0..n)] # Peter Luschny, Sep 17 2014 and Mar 16 2016

# Alternatively:

def A113413_row(n):

    @cached_function

    def prec(n, k):

        if k==n: return 1

        if k==0: return 0

        return prec(n-1, k-1)+2*sum(prec(n-i, k-1) for i in (2..n-k+1))

    return [prec(n, k) for k in (1..n)]

for n in (1..10): print A113413_row(n) # Peter Luschny, Mar 16 2016

CROSSREFS

Other versions: A035607, A119800, A122542, A266213.

Sequence in context: A208755 A226441 A080246 * A125694 A136678 A110162

Adjacent sequences:  A113410 A113411 A113412 * A113414 A113415 A113416

KEYWORD

easy,nonn,tabl,look

AUTHOR

Paul Barry, Oct 29 2005

STATUS

approved

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Last modified September 20 16:14 EDT 2017. Contains 292276 sequences.