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 A033184 Catalan triangle A009766 transposed. 87
 1, 1, 1, 2, 2, 1, 5, 5, 3, 1, 14, 14, 9, 4, 1, 42, 42, 28, 14, 5, 1, 132, 132, 90, 48, 20, 6, 1, 429, 429, 297, 165, 75, 27, 7, 1, 1430, 1430, 1001, 572, 275, 110, 35, 8, 1, 4862, 4862, 3432, 2002, 1001, 429, 154, 44, 9, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Triangle read by rows: T(n,k) = number of Dyck n-paths (A000108) containing k returns to ground level. E.g., the paths UDUUDD, UUDDUD each have 2 returns; so T(3,2)=2. Row sums over even-indexed columns are the Fine numbers A000957. - David Callan, Jul 25 2005 Triangular array of numbers a(n,k) = number of linear forests of k planted planar trees and n non-root nodes. Catalan convolution triangle; with offset [0,0]: a(n,m)=(m+1)*binomial(2*n-m,n-m)/(n+1), n >= m >= 0, else 0. G.f. for column m: c(x)*(x*c(x))^m with c(x) g.f. for A000108 (Catalan). - Wolfdieter Lang, Sep 12 2001 a(n+1,m+1), n >= m >= 0, a(n,m) := 0, n= m > 1. - Vladimir Kruchinin, Mar 17 2011 T(n-1,m-1) = (m/n)*Sum_{k=1..n-m+1} (k*A000108(k-1)*T(n-k-1,m-2)), n >= m > 1. - Vladimir Kruchinin, Mar 17 2011 T(n,k) = C(2*n-k-1,n-k) - C(2*n-k-1,n-k-1). - Dennis P. Walsh, Mar 19 2012 T(n,k) = C(2*n-k,n)*k/(2*n-k). - Dennis P. Walsh, Mar 19 2012 T(n,k) = T(n,k-1) - T(n-1,k-2). - Dennis P. Walsh, Mar 19 2012 G.f.: 2*x*y / (1 + sqrt(1 - 4*x) - 2*x*y) = Sum_{n >= k > 0} T(n, k) * x^n * y^k. - Michael Somos, Jun 06 2016 EXAMPLE Triangle begins:   ---+-----------------------------------   n\k|   1    2    3    4    5    6    7   ---+-----------------------------------    1 |   1    2 |   1    1    3 |   2    2    1    4 |   5    5    3    1    5 |  14   14    9    4    1    6 |  42   42   28   14    5    1    7 | 132  132   90   48   20    6    1 MAPLE a := proc(n, k) if k<=n then k*binomial(2*n-k, n)/(2*n-k) else 0 fi end: seq(seq(a(n, k), k=1..n), n=1..10); MATHEMATICA nn = 10; c = (1 - (1 - 4 x)^(1/2))/(2 x); f[list_] := Select[list, # > 0 &]; Map[f, Drop[CoefficientList[Series[y x c/(1 - y x c), {x, 0, nn}], {x, y}], 1]] //Flatten (* Geoffrey Critzer, Jan 31 2012 *) Flatten[Reverse /@ NestList[Append[Accumulate[#], Last[Accumulate[#]]] &, {1}, 9]] (* Birkas Gyorgy, May 19 2012 *) PROG (PARI) T(n, k)=binomial(2*(n-k)+k, n-k)*(k+1)/(n+1) \\ Paul D. Hanna, Aug 11 2008 (Sage) # The simplest way to construct the triangle. def A033184_triangle(n) :     T = [0 for i in (0..n)]     for k in (1..n) :         T[k] = 1         for i in range(k-1, 0, -1) :             T[i] = T[i-1] + T[i+1]         print([T[i] for i in (1..k)]) A033184_triangle(10) # Peter Luschny, Jan 27 2012 (Haskell) a033184 n k = a033184_tabl !! (n-1) !! (k-1) a033184_row n = a033184_tabl !! (n-1) a033184_tabl = map reverse a009766_tabl -- Reinhard Zumkeller, Feb 19 2014 (Magma) /* As triangle: */ [[Binomial(2*n-k, n)*k/(2*n-k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Oct 12 2015 CROSSREFS Rows of Catalan triangle A009766 read backwards. a(n, 1) = A000108(n-1). Row sums = A000108(n) (Catalan). The following are all versions of (essentially) the same Catalan triangle: A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072. Diagonals give A000108 A000245 A002057 A000344 A003517 A000588 A003518 A003519 A001392, ... Cf. A116364 (row squared sums). - Paul D. Hanna, Aug 11 2008 Sequence in context: A190252 A141751 A079222 * A171567 A110488 A271025 Adjacent sequences:  A033181 A033182 A033183 * A033185 A033186 A033187 KEYWORD nonn,tabl AUTHOR STATUS approved

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Last modified July 1 08:17 EDT 2022. Contains 354953 sequences. (Running on oeis4.)