%I #58 May 12 2024 23:35:18
%S 1,1,1,1,4,1,1,8,8,1,1,13,29,13,1,1,19,73,73,19,1,1,26,151,266,151,26,
%T 1,1,34,276,749,749,276,34,1,1,43,463,1781,2762,1781,463,43,1,1,53,
%U 729,3758,8321,8321,3758,729,53,1,1,64,1093,7253,21659,31004,21659,7253,1093,64,1
%N Triangle read by rows: T(n, k) = number of hill-free Dyck paths (i.e., no peaks at height 1) of semilength n and having k peaks.
%C Row n has n - 1 terms. Row sums yield the Fine numbers (A000957).
%C Related to the number of certain sets of non-crossing partitions for the root system A_n (p. 11, Athanasiadis and Savvidou). - _Tom Copeland_, Oct 19 2014
%C T(n,k) is the number of permutations pi of [n-1] with k - 1 descents such that s(pi) avoids the patterns 132, 231, and 312, where s is West's stack-sorting map. - _Colin Defant_, Sep 16 2018
%C The absolute values of the polynomials at -1 and j (cube root of 1) seem to be given by A126120 and A005043. - _F. Chapoton_, Nov 16 2021
%C _Don Knuth_ observes that this sequence also arrises from the enumeration of restricted max-and-min-closed relations, only there it appears as an array read by antidiagonals: see the Knuth "Notes" link and A372068. Knuth also gives a formula expressing the array A372368 in terms of this array. He also reports that there is strong experimental evidence that the n-th term of row m in this array is a polynomial of degree 2*m-2 in n. - _N. J. A. Sloane_, May 12 2024
%H Alois P. Heinz, <a href="/A100754/b100754.txt">Rows n = 2..142, flattened</a>
%H C. Athanasiadis and C. Savvidou, <a href="http://arxiv.org/abs/1204.0362">The local h-vector of the cluster subdivision of a simplex</a>, arXiv preprint arXiv:1204.0362 [math.CO], 2012.
%H Paul Barry, <a href="https://arxiv.org/abs/1803.06408">Three Études on a sequence transformation pipeline</a>, arXiv:1803.06408 [math.CO], 2018.
%H Colin Defant, <a href="https://arxiv.org/abs/1809.03123">Stack-sorting preimages of permutation classes</a>, arXiv:1809.03123 [math.CO], 2018.
%H E. Deutsch and L. Shapiro, <a href="http://dx.doi.org/10.1016/S0012-365X(01)00121-2">A survey of the Fine numbers</a>, Discrete Math., 241 (2001), 241-265.
%H D. E. Knuth, <a href="/A372066/a372066.txt">Notes on four arrays of numbers arising from the enumeration of CRC constraints and min-and-max-closed constraints</a>, May 06 2024
%F T(n, k) = Sum_{j=0..min(k, n-k)} (j/(n-j)) * C(n-j, k-j) * C(n-j, k), n >= 2.
%F G.f.: t*z*r/(1 - t*z*r), where r = r(t, z) is the Narayana function defined by r = z*(1 + r)*(1 + t*r).
%F From _Tom Copeland_, Oct 19 2014: (Start)
%F With offset 0 for A108263 and offset 1 for A132081, row polynomials of this entry P(n, x) = Sum_{i} A108263(n, i)*x^i*(1 + x)^(n - 2*i)) = Sum_{i} A132081(n - 2, i)*x^i*(1 + x)^(n - 2*i)).
%F E.g., P(4, x) = 1*x*(1 + x)^(4 - 2*1) + 2*x^2*(1 + x)^(4 - 2*2) = x + 4 x^2 + x^3.
%F Equivalently, let Q(n, x) be the row polynomials of A108263. Then P(n, x) = (1 + x)^n * Q(n, x/(1 + x)^2).
%F E.g., P(4, x) = (1 + x)^4 [x/(1 + x)^2 + 2 [x/(1 + x)^2)^2]].
%F See Athanasiadis and Savvidou (p. 7). (End)
%e T(4, 2) = 4 because we have UU*DDUU*DD, UU*DUU*DDD, UUU*DDU*DD and UUU*DU*DDD, where U = (1, 1), D = (1,-1) and * indicates the peaks.
%e Triangle starts:
%e 1;
%e 1, 1;
%e 1, 4, 1;
%e 1, 8, 8, 1;
%e 1, 13, 29, 13, 1;
%e 1, 19, 73, 73, 19, 1;
%e 1, 26, 151, 266, 151, 26, 1;
%e 1, 34, 276, 749, 749, 276, 34, 1;
%e 1, 43, 463, 1781, 2762, 1781, 463, 43, 1;
%e 1, 53, 729, 3758, 8321, 8321, 3758, 729, 53, 1;
%e ...
%e As an array (for which the rows of the preceding triangle are the antidiagonals):
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 4, 8, 13, 19, 26, 34, 43, 53, ...
%e 1, 8, 29, 73, 151, 276, 463, 729, 1093, ...
%e 1, 13, 73, 266, 749, 1781, 3758, 7253, 13061, ...
%e 1, 19, 151, 749, 2762, 8321, 21659, 50471, 107833, ...
%e 1, 26, 276, 1781, 8321, 31004, 97754, 271125, 679355, ...
%e 1, 34, 463, 3758, 21659, 97754, 367285, 1196665, 3478915, ...
%e 1, 43, 729, 7253, 50471, 271125, 1196665, 4526470, 15118415, ...
%e 1, 53, 1093, 13061, 107833, 679355, 3478915, 15118415, 57500480, ...
%e ...
%p T := (n, k) -> add((j/(n-j))*binomial(n-j, k-j)*binomial(n-j,k), j=0..min(k,n-k)): for n from 2 to 13 do seq(T(n, k), k = 1..n-1) od; # yields the sequence in triangular form
%t T[n_, k_] := Sum[(j/(n-j))*Binomial[n-j, k-j]*Binomial[n-j, k], {j, 0, Min[k, n-k]}]; Table[T[n, k], {n, 2, 13}, {k, 1, n-1}] // Flatten (* _Jean-François Alcover_, Feb 19 2017, translated from Maple *)
%Y Cf. A000957, A108263, A132081.
%Y See also A099594, A272644, A372066, A372067, A372068.
%K nonn,tabl
%O 2,5
%A _Emeric Deutsch_, Jan 14 2005.