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 A100754 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. 10
 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, 1, 64, 1093, 7253, 21659, 31004, 21659, 7253, 1093, 64, 1 (list; table; graph; refs; listen; history; text; internal format)
 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 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/(n-j)*C(n-j,k-j)*C(n-j,k), j=0..min(k, n-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(over i, A108263(n,i)*x^i*(1+x)^(n-2*i)) = sum(over 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; MAPLE T:=(n, k)->sum((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 Cf. A000957. Sequence in context: A177947 A132789 A319251 * A296405 A174035 A303888 Adjacent sequences:  A100751 A100752 A100753 * A100755 A100756 A100757 KEYWORD nonn,tabl AUTHOR Emeric Deutsch, Jan 14 2005 STATUS approved

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Last modified October 19 22:28 EDT 2018. Contains 316378 sequences. (Running on oeis4.)