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 A145596 Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). 8
 1, 2, 2, 3, 8, 3, 4, 20, 20, 4, 5, 40, 75, 40, 5, 6, 70, 210, 210, 70, 6, 7, 112, 490, 784, 490, 112, 7, 8, 168, 1008, 2352, 2352, 1008, 168, 8, 9, 240, 1890, 6048, 8820, 6048, 1890, 240, 9, 10, 330, 3300, 13860, 27720, 27720, 13860, 3300, 330, 10 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS T(n,k) is the number of walks of n unit steps on the square lattice (i.e., each step in the direction either up (U), down (D), right (R) or left (L)) starting from (0,0) and finishing at points on the horizontal line y = 1, which remain in the upper half-plane y >= 0. An example is given in the Example section below. The current array is the case r = 1 of the generalized Narayana numbers N_r(n, k) := (r + 1)/(n + 1)*binomial(n + 1, k + r)*binomial(n + 1, k - 1), which count walks of n steps from the origin to points on the horizontal line y = r that remain in the upper half-plane. Case r = 0 gives the table of Narayana numbers A001263 (but with row numbering starting at n = 0). For other cases see A145597 (r = 2), A145598 (r = 3) and A145599 (r = 4). T(n,k) is the number of preimages of the permutation 21345...(n+2) under West's stack-sorting map that have exactly k descents. - Colin Defant, Sep 15 2018 LINKS F. Cai, Q.-H. Hou, Y. Sun, A. L. B. Yang, Combinatorial identities related to 2x2 submatrices of recursive matrices, arXiv:1808.05736 [math.CO],2018; Table 2.1 for k=1. C. Defant, Preimages under the stack-sorting algorithm, arXiv:1511.05681 [math.CO], 2015-2018; Graphs Combin., 33 (2017), 103-122. C. Defant, Stack-sorting preimages of permutation classes, arXiv:1809.03123 [math.CO], 2018. R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6. T. Mansour, Y. Sun, Identities involving Narayana polynomials and Catalan numbers, arXiv:0805.1274 [math.CO], 2008. FORMULA T(n,k) = 2/(n + 1)*binomial(n + 1,k + 1)*binomial(n + 1,k - 1) for 1 <= k <= n. In the notation of [Guy], T(n,k) equals w_n(x,y) at (x,y) = (2*k - n - 1,1). O.g.f. for column (k + 2): 2/(k + 1) * y^(k+2)/(1 - y)^(k+4) * Jacobi_P(k,2,1,(1 + y)/(1 - y)). The column generating functions begin: column 2: 2*y^2/(1 - y)^4; column 3: y^3*(3 + 2*y)/(1 - y)^6; column 4: y^4*(4 + 8*y + 2*y^2)/(1 - y)^8; the polynomials in the numerators are the row generating polynomials of array A108838. O.g.f. for array: 1/(2*x*y^3) * (((1 + x)*y - 1)*sqrt(1 - 2*(1 + x)*y + (y - x*y)^2) + x^2*y^2 - 2*x*y + (1 - y)^2) = x*y + (2*x + 2*x^2)*y^2 + (3*x + 8*x^2 + 3*x^3)*y^3 + (4*x + 20*x^2 + 20*x^3 + 4*x^4)*y^4 + .... Row sums A002057. Identities for row polynomials R_n(x) = Sum_{k = 1..n} T(n,k)*x^k (compare with the results in section 1 of [Mansour & Sun]): x*R_(n-1)(x) = 2*(n - 1)/((n + 1)*(n + 2)) * Sum_{k = 0..n} binomial(n + 2,k) * binomial(2*n - k,n) * (x - 1)^k; R_n(x) = Sum_{k = 0..floor((n-1)/2)} binomial(n, 2*k + 1) * Catalan(k + 1) * x^(k+1)*(1 + x)^(n-2k-1); Sum {k = 1..n} (-1)^(n-k)*binomial(n,k)*R_k(x)*(1 + x)^(n-k) = x^m*Catalan(m) if n = 2*m - 1 is odd, otherwise the sum is zero. Sum {k = 1..n} (-1)^(k+1)*binomial(n,k)*R_k(x^2)*(1 + x)^(2*(n-k)) = R_n(1)*x^(n+1) = 4/(n + 3)*binomial(2*n + 1,n - 1)*x^(n+1) = A002057(n-1)*x^(n+1). Row generating polynomial R_(n+1)(x) = 2/(n + 2)*x*(1 - x)^n * Jacobi_P(n,2,2,(1 + x)/(1 - x)). - Peter Bala, Oct 31 2008 G.f. satisfies x^3*y*A(x,y)^2-A(x,y)*(x^2*y^2+(-2)*x*y+x^2+(-2)*x+1)+x=0. - Vladimir Kruchinin, Oct 11 2020 EXAMPLE n\k|..1.....2....3.....4.....5.....6 ==================================== .1.|..1 .2.|..2.....2 .3.|..3.....8....3 .4.|..4....20...20.....4 .5.|..5....40...75....40.....5 .6.|..6....70..210...210....70.....6 ... Row 3 entries: T(3,1) = 3: the 3 walks from (0,0) to (-2,1) of three steps are LLU, LUL and ULL. T(3,2) = 8: the 8 walks from (0,0) to (0,1) of three steps are UDU, UUD, ULR, URL, RLU, LRU, RUL and LUR. T(3,3) = 3: the 3 walks from (0,0) to (2,1) of three steps are RRU, RUR and URR. . . *......*......*......y......*......*......* . . *......3......*......8......*......3......* . . *......*......*......o......*......*......* x-axis . . MAPLE T:= (n, k) -> 2/(n+1)*binomial(n+1, k+1)*binomial(n+1, k-1): for n from 1 to 10 do seq(T(n, k), k = 1..n) end do; MATHEMATICA t[n_, k_]:=2/(n+1) Binomial[n+1, k+1] Binomial[n+1, k-1]; Table[t[n, k], {n, 3, 10}, {k, n}]//Flatten (* Vincenzo Librandi, Sep 15 2018 *) PROG (MAGMA) /* As triangle */  [[2/(n+1)*Binomial(n+1, k+1)*Binomial(n+1, k-1): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Sep 15 2018 CROSSREFS Cf. A002057 (row sums), A001263, A108838, A145597, A145598, A145599, A145600. Sequence in context: A296952 A141611 A234357 * A186753 A135835 A177696 Adjacent sequences:  A145593 A145594 A145595 * A145597 A145598 A145599 KEYWORD easy,nonn,tabl AUTHOR Peter Bala, Oct 14 2008 STATUS approved

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Last modified April 15 06:12 EDT 2021. Contains 342975 sequences. (Running on oeis4.)