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 A145597 Generalized Narayana numbers, T(n, k) = 3/(n + 1)*binomial(n + 1, k + 2)*binomial(n + 1, k - 1), triangular array read by rows. 5
 1, 3, 3, 6, 15, 6, 10, 45, 45, 10, 15, 105, 189, 105, 15, 21, 210, 588, 588, 210, 21, 28, 378, 1512, 2352, 1512, 378, 28, 36, 630, 3402, 7560, 7560, 3402, 630, 36, 45, 990, 6930, 20790, 29700, 20790, 6930, 990, 45, 55, 1485, 13068, 50820, 98010, 98010 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 2,2 COMMENTS T(n,k) is the number of walks of n unit steps, each step in the direction either up (U), down (D), right (R) or left (L), starting from (0,0) and finishing at lattice points on the horizontal line y = 2 and 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 = 2 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 an offset of 0 in the row numbering). For other cases see A145596 (r = 1), A145598 (r = 3) and A145599 (r = 4). T(n,k) is the number of preimages of the permutation 3214567...(n+3) under West's stack-sorting map that have exactly k+1 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=2. C. Defant, Preimages under the stack-sorting algorithm, 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. FORMULA T(n,k) = 3/(n+1)*binomial(n+1,k+2)*binomial(n+1,k-1) for n >=2 and 1 <= k <= n-1. In the notation of [Guy], T(n,k) equals w_n(x,y) at (x,y) = (2*k - n,2). Row sums A003517. O.g.f. for column k+2: 3/(k + 1) * y^(k+3)/(1 - y)^(k+5) * Jacobi_P(k,3,1,(1 + y)/(1 - y)). Identities for row polynomials R_n(x) := sum {k = 1..n-1} T(n,k)*x^k: x^2*R_(n-1)(x) = 3*(n-1)*(n-2)/((n+1)*(n+2)*(n+3)) * sum {k = 0..n} binomial(n + 3,k) * binomial(2n - k,n) * (x - 1)^k; sum {k = 1..n} (-1)^k*binomial(n,k)*R_k(x^2)*(1 + x)^(2*(n-k)) = R_n(1)*x^n = 6/(n+4)*binomial(2n+1,n-2)*x^n = A003517(n)*x^n. Row generating polynomial R_(n+2)(x) = 3/(n+3)*x*(1-x)^n * Jacobi_P(n,3,3,(1+x)/(1-x)). [Peter Bala, Oct 31 2008] EXAMPLE Triangle starts n\k|..1.....2....3.....4.....5.....6 ==================================== .2.|..1 .3.|..3.....3 .4.|..6....15....6 .5.|.10....45...45....10 .6.|.15...105..189...105....15 .7.|.21...210..588...588...210....21 ... Row 4: T(4,1) = 6: the 6 walks of length 4 from (0,0) to (-2,2) are LLUU, LULU, LUUL, ULLU, ULUL and UULL. Changing L to R in these walks gives the 6 walks from (0,0) to (2,2). T(4,2) = 15: the 15 walks of length 4 from (0,0) to (0,2) are UUUD, UULR, UURL, UUDU,URUL, ULUR, URLU, ULRU, RUUL, LUUR, RLUU, LRUU, RULU, LURU and UDUU. . . *......*......*......y......*......*......* . . *......6......*.....15......*......6......* . . *......*......*......*......*......*......* . . *......*......*......o......*......*......* x axis . MAPLE with(combinat): T:= (n, k) -> 3/(n+1)*binomial(n+1, k+2)*binomial(n+1, k-1): for n from 2 to 11 do seq(T(n, k), k = 1..n-1); end do; CROSSREFS Cf. A003517 (row sums), A001263, A145596, A145598, A145599, A145601. Sequence in context: A096572 A318540 A110523 * A143418 A092370 A245796 Adjacent sequences:  A145594 A145595 A145596 * A145598 A145599 A145600 KEYWORD easy,nonn,tabl AUTHOR Peter Bala, Oct 15 2008 STATUS approved

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Last modified October 23 12:10 EDT 2018. Contains 316527 sequences. (Running on oeis4.)