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%I #73 Feb 28 2023 17:01:48
%S 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,
%T 24,102,192,170,72,14,1,2,28,146,360,450,292,98,16,1,2,32,198,608,
%U 1002,912,462,128,18,1,2,36,258,952,1970,2364,1666,688,162,20,1,2,40,326
%N A Riordan array of coordination sequences.
%C 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.
%C For another version see A122542. - _Philippe Deléham_, Oct 15 2006
%C 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
%C From _Eric W. Weisstein_, Feb 17 2016: (Start)
%C Triangle of coefficients (from low to high degree) of x^-n * vertex cover polynomial of the n-ladder graph P_2 \square p_n:
%C Psi_{L_1}: x*(2 + x) -> {2, 1}
%C Psi_{L_2}: x^2*(2 + 4 x + x^2) -> {2, 4, 1}
%C Psi_{L_3}: x^3*(2 + 8 x + 6 x^2 + x^3) -> {2, 8, 6, 1}
%C (End)
%C 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
%H G. C. Greubel, <a href="/A113413/b113413.txt">Table of n, a(n) for the first 50 rows</a>
%H Bela Bajnok, <a href="https://arxiv.org/abs/1705.07444">Additive Combinatorics: A Menu of Research Problems</a>, arXiv:1705.07444 [math.NT], May 2017. See Sect. 2.3.
%H Paul Barry, <a href="http://dx.doi.org/10.1016/j.laa.2015.10.032">Riordan arrays, generalized Narayana triangles, and series reversion</a>, Linear Algebra and its Applications, 491 (2016) 343-385.
%H P. Holub, M. Miller, H. Perez-Roses, and J. Ryan, <a href="https://doi.org/10.1016/j.dam.2014.07.012">Degree diameter problem on honeycomb networks</a>, Disc. Appl. Math. 179 (2014) 139-151.
%H Milan Janjić, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Janjic/janjic93.html">Words and Linear Recurrences</a>, J. Int. Seq. 21 (2018), #18.1.4.
%H Huyile Liang, Yanni Pei, and Yi Wang, <a href="https://arxiv.org/abs/2302.11856">Analytic combinatorics of coordination numbers of cubic lattices</a>, arXiv:2302.11856 [math.CO], 2023. See p. 2.
%H Mirka Miller, Hebert Perez-Roses, and Joe Ryan, <a href="http://arxiv.org/abs/1203.4069">The Maximum Degree-and-Diameter-Bounded Subgraph in the Mesh</a>, arXiv preprint arXiv:1203.4069 [math.CO], 2012.
%F From _Paul Barry_, Nov 13 2005: (Start)
%F Riordan array ((1+x)/(1-x), x(1+x)/(1-x)).
%F T(n, k) = Sum_{i=0..n-k} C(k+1, i)*C(n-i, k).
%F T(n, k) = Sum_{j=0..n-k} C(k+j, j)*C(k+1, n-k-j).
%F 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).
%F T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-2, k-1).
%F T(n, k) = Sum_{j=0..n} C(floor((n+j)/2), k)*C(k, floor((n-j)/2)). (End)
%F T(n, k) = C(n, k)*hypergeometric([-k-1, k-n], [-n], -1). - _Peter Luschny_, Sep 17 2014
%F 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
%e Triangle begins
%e 1;
%e 2, 1;
%e 2, 4, 1;
%e 2, 8, 6, 1;
%e 2, 12, 18, 8, 1;
%e 2, 16, 38, 32, 10, 1;
%e 2, 20, 66, 88, 50, 12, 1;
%t 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 *)
%t 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 *)
%o (Sage)
%o T = lambda n,k : binomial(n, k)*hypergeometric([-k-1, k-n], [-n], -1).simplify_hypergeometric()
%o A113413 = lambda n,k : 1 if n==0 and k==0 else T(n, k)
%o for n in (0..12): print([A113413(n,k) for k in (0..n)]) # _Peter Luschny_, Sep 17 2014 and Mar 16 2016
%o (Sage)
%o # Alternatively:
%o def A113413_row(n):
%o @cached_function
%o def prec(n, k):
%o if k==n: return 1
%o if k==0: return 0
%o return prec(n-1,k-1)+2*sum(prec(n-i,k-1) for i in (2..n-k+1))
%o return [prec(n, k) for k in (1..n)]
%o for n in (1..10): print(A113413_row(n)) # _Peter Luschny_, Mar 16 2016
%Y Other versions: A035607, A119800, A122542, A266213.
%K easy,nonn,tabl,look
%O 0,2
%A _Paul Barry_, Oct 29 2005
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