A113413 A Riordan array of coordination sequences.

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

Offset: 0

Author

Paul Barry, Oct 29 2005

Keywords

• easy
• nonn
• tabl
• look

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)

Let c(n, k), n > 0, be multiplicative sequences for some fixed integer k >= 0 with c(p^e, k) = T(e+k, k) for prime p and e >= 0. Then we have Dirichlet g.f.: Sum_{n>0} c(n, k) / n^s = zeta(s)^(2*k+2) / zeta(2*s)^(k+1). Examples: For k = 0 see A034444 and for k = 1 see A322328. Dirichlet convolution of c(n, k) and lambda(n) is Dirichlet inverse of c(n, k). - Werner Schulte, Oct 31 2022

Examples

			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;
		

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.
• P. Holub, M. Miller, H. Perez-Roses, and J. Ryan, Degree diameter problem on honeycomb networks, Disc. Appl. Math. 179 (2014) 139-151.
• Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.
• Huyile Liang, Yanni Pei, and Yi Wang, Analytic combinatorics of coordination numbers of cubic lattices, arXiv:2302.11856 [math.CO], 2023. See p. 2.
• 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.

Crossrefs

Other versions: A035607, A119800, A122542, A266213.

Programs

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 *)

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

Sage

# 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

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

T(n, k) = (Sum_{i=2..k+2} A137513(k+2, i) * (n-k)^(i-2)) / (k!) for 0 <= k < n (conjectured). - Werner Schulte, Oct 31 2022