OFFSET
2,5
COMMENTS
Row n has n - 1 terms. Row sums yield the Fine numbers (A000957).
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
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
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
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
LINKS
Alois P. Heinz, Rows n = 2..142, flattened
C. Athanasiadis and C. Savvidou, The local h-vector of the cluster subdivision of a simplex, arXiv preprint arXiv:1204.0362 [math.CO], 2012.
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Colin Defant, Stack-sorting preimages of permutation classes, arXiv:1809.03123 [math.CO], 2018.
E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
FORMULA
T(n, k) = Sum_{j=0..min(k, n-k)} (j/(n-j)) * C(n-j, k-j) * C(n-j, k), n >= 2.
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).
From Tom Copeland, Oct 19 2014: (Start)
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)).
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.
Equivalently, let Q(n, x) be the row polynomials of A108263. Then P(n, x) = (1 + x)^n * Q(n, x/(1 + x)^2).
E.g., P(4, x) = (1 + x)^4 [x/(1 + x)^2 + 2 [x/(1 + x)^2)^2]].
See Athanasiadis and Savvidou (p. 7). (End)
EXAMPLE
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.
Triangle starts:
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, 1;
1, 34, 276, 749, 749, 276, 34, 1;
1, 43, 463, 1781, 2762, 1781, 463, 43, 1;
1, 53, 729, 3758, 8321, 8321, 3758, 729, 53, 1;
...
As an array (for which the rows of the preceding triangle are the antidiagonals):
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 4, 8, 13, 19, 26, 34, 43, 53, ...
1, 8, 29, 73, 151, 276, 463, 729, 1093, ...
1, 13, 73, 266, 749, 1781, 3758, 7253, 13061, ...
1, 19, 151, 749, 2762, 8321, 21659, 50471, 107833, ...
1, 26, 276, 1781, 8321, 31004, 97754, 271125, 679355, ...
1, 34, 463, 3758, 21659, 97754, 367285, 1196665, 3478915, ...
1, 43, 729, 7253, 50471, 271125, 1196665, 4526470, 15118415, ...
1, 53, 1093, 13061, 107833, 679355, 3478915, 15118415, 57500480, ...
...
MAPLE
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
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Jan 14 2005.
STATUS
approved