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%I #25 Sep 19 2019 07:57:32
%S 1,1,1,1,2,1,1,3,5,1,1,4,11,13,1,1,5,19,41,34,1,1,6,29,91,153,89,1,1,
%T 7,41,169,436,571,233,1,1,8,55,281,985,2089,2131,610,1,1,9,71,433,
%U 1926,5741,10009,7953,1597,1,1,10,89,631,3409,13201,33461,47956,29681
%N Array T(k,n) read by antidiagonals. G.f.: x(1-x)/(1-kx+x^2), k>1.
%C Also, values of polynomials with coefficients in A098493 (see Fink et al.). See A098495 for negative k.
%C Number of dimer tilings of the graph S_{k-1} X P_{2n-2}.
%H Alex Fink, Richard K. Guy, and Mark Krusemeyer, <a href="https://doi.org/10.11575/cdm.v3i2.61940">Partitions with parts occurring at most thrice</a>, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114.
%H Elizabeth Wilmer, <a href="http://www.oberlin.edu/math/faculty/wilmer/OEISconj87.pdf">A note on Stephan's conjecture 87</a>
%H Elizabeth Wilmer, <a href="/A094954/a094954.pdf">A note on Stephan's conjecture 87</a> [cached copy]
%F Recurrence: T(k, 1) = 1, T(k, 2) = k-1, T(k, n) = kT(k, n-1) - T(k, n-2).
%F For n>3, T(k, n) = [k(k-2) + T(k, n-1)T(k, n-2)] / T(k, n-3).
%F T(k, n+1) = S(n, k) - S(n-1, k) = U(n, k/2) - U(n-1, k/2), with S, U = Chebyshev polynomials of second kind.
%F T(k+2, n+1) = Sum[i=0..n, k^(n-i) * C(2n-i, i)] (from comments by Benoit Cloitre).
%e 1,1,1,1,1,1,1,1,1,1,1,1,1,1, ...
%e 1,2,5,13,34,89,233,610,1597, ...
%e 1,3,11,41,153,571,2131,7953, ...
%e 1,4,19,91,436,2089,10009,47956, ...
%e 1,5,29,169,985,5741,33461,195025, ...
%e 1,6,41,281,1926,13201,90481,620166, ...
%t max = 14; row[k_] := Rest[ CoefficientList[ Series[ x*(1-x)/(1-k*x+x^2), {x, 0, max}], x]]; t = Table[ row[k], {k, 2, max+1}]; Flatten[ Table[ t[[k-n+1, n]], {k, 1, max}, {n, 1, k}]] (* _Jean-François Alcover_, Dec 27 2011 *)
%o (PARI) T(k,n)=polcoeff(x*(1-x)/(1-k*x+x*x),n)
%Y Rows are first differences of rows in array A073134.
%Y Rows 2-14 are A000012, A001519, A079935/A001835, A004253, A001653, A049685, A070997, A070998, A072256, A078922, A077417, A085260, A001570. Other rows: A007805 (k=18), A075839 (k=20), A077420 (k=34), A078988 (k=66).
%Y Columns include A028387. Diagonals include A094955, A094956. Antidiagonal sums are A094957.
%K nonn,tabl
%O 1,5
%A _Ralf Stephan_, May 31 2004
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