The OEIS is supported by the many generous donors to the OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A100100 Triangle T(n,k) = binomial(2*n-k-1, n-k) read by rows. 22
 1, 1, 1, 3, 2, 1, 10, 6, 3, 1, 35, 20, 10, 4, 1, 126, 70, 35, 15, 5, 1, 462, 252, 126, 56, 21, 6, 1, 1716, 924, 462, 210, 84, 28, 7, 1, 6435, 3432, 1716, 792, 330, 120, 36, 8, 1, 24310, 12870, 6435, 3003, 1287, 495, 165, 45, 9, 1, 92378, 48620, 24310, 11440, 5005, 2002 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Sometimes called a Catalan triangle, although there are many other triangles that carry that name - see A009766, A008315, A028364, A033184, A053121, A059365, A062103. Number of nodes of outdegree k in all ordered trees with n edges. Equivalently, number of ascents of length k in all Dyck paths of semilength n. Example: T(3,2) = 3 because the Dyck paths of semilength 3 are UDUDUD, UD(UU)DD, (UU)DDUD, (UU)DUDD and UUUDDD, where U = (1,1), D = (1,-1), the ascents of length 2 being shown between parentheses. - Emeric Deutsch, Nov 19 2006 Riordan array (f(x), x*g(x)) where f(x) is the g.f. of A088218 and g(x) is the g.f. of A000108. - Philippe Deléham, Jan 23 2010 T(n,k) is the number of nonnegative paths of upsteps U = (1,1) and downsteps D = (1,-1) of length 2*n with k returns to  ground level, the horizontal line through the initial vertex. Example: T(2,1) = 2 counts UDUU, UUDD. Also, T(n,k) = number of these paths whose last descent has length k, that is, k downsteps follow the last upstep. Example: T(2,1) = 2 counts UUUD, UDUD. - David Callan, Nov 21 2011 Belongs to the hitting-time subgroup of the Riordan group. Multiplying this triangle by the square Pascal matrix gives A092392 read as a square array. See the example below. - Peter Bala, Nov 03 2015 LINKS Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5. Gi-Sang Cheon, Hana Kim, and Louis W. Shapiro, An algebraic structure for Faber polynomials, Lin. Alg. Applic. 433 (2010) 1170-1179. Tian-Xiao He and Renzo Sprugnoli, Sequence characterization of Riordan arrays, Discrete Math. 309 (2009), no. 12, 3962-3974. FORMULA From Peter Bala, Sep 06 2015: (Start) Matrix product A094527 * P^(-1) = A113955 * P^(-2), where P denotes Pascal's triangle A007318. Essentially, the logarithmic derivative of A033184. (End) Let column(k) = [T(n, k), n >= k], then the generating function for column(k) is (2/(sqrt(1-4*x)+1))^(k-1)/sqrt(1-4*x). - Peter Luschny, Mar 19 2021 O.g.f. row polynomials R(n, x) := Sum_{k=0..n} T(n, k)*x^k, i.e. o.g.f. of the triangle, G(z,x) = 1/((2 - c(z))*(1 - x*z*c(z))), with c the o.g.f. of A000108 (Catalan). See the Riordan coomment by Philippe Deléham above. - Wolfdieter Lang, Apr 06 2021 EXAMPLE From Paul Barry, Mar 15 2010: (Start) Triangle begins in row n=0 with columns 0<=k<=n as:     1;     1,   1;     3,   2,   1;    10,   6,   3,  1;    35,  20,  10,  4,  1;   126,  70,  35, 15,  5, 1;   462, 252, 126, 56, 21, 6, 1; Production matrix begins   1, 1;   2, 1, 1;   3, 1, 1, 1;   4, 1, 1, 1, 1;   5, 1, 1, 1, 1, 1;   6, 1, 1, 1, 1, 1, 1;   7, 1, 1, 1, 1, 1, 1, 1; (End) A092392 as a square array = A100100 * square Pascal matrix: /1   1  1  1 ...\   / 1          \/1 1  1  1 ...\ |2   3  4  5 ...|   | 1 1        ||1 2  3  4 ...| |6  10 15 21 ...| = | 3 2 1      ||1 3  6 10 ...| |20 35 56 84 ...|   |10 6 3 1    ||1 4 10 20 ...| |70 ...         |   |35 ...      ||1 ...        | - Peter Bala, Nov 03 2015 MAPLE A100100 := proc(n, k)     binomial(2*n-k-1, n-1) ; end proc: seq(seq(A100100(n, k), k=0..n), n=0..10) ; # R. J. Mathar, Feb 06 2015 MATHEMATICA Flatten[Table[Binomial[2 n - k - 1, n - k], {n, 0, 11}, {k, 0, n}]] (* Vincenzo Librandi, Nov 21 2018 *) PROG (PARI) T(n, k)=binomial(2*n-k-1, n-k) \\ Charles R Greathouse IV, Jan 16 2012 (Haskell) a100100 n k = a100100_tabl !! n !! n a100100_row n = a100100_tabl !! n a100100_tabl =  : f a092392_tabl where    f (us : wss'@(vs : wss)) = (vs !! 1 : us) : f wss' -- Reinhard Zumkeller, Jan 15 2014 (MAGMA) /* As triangle */ [[Binomial(2*n - k - 1, n - k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Nov 21 2018 CROSSREFS Row sums are A000984. Equivalent to A092392, to which A088218 has been added as a first column. Columns include A088218, A000984, A001700, A001791, A002054, A002694. Diagonal sums are A100217. Matrix inverse is A100218. Cf. A059481 (mirrored). Cf. A033184, A094527, A113955. Sequence in context: A267742 A267278 A267019 * A248036 A307214 A185967 Adjacent sequences:  A100097 A100098 A100099 * A100101 A100102 A100103 KEYWORD easy,nonn,tabl AUTHOR Paul Barry, Nov 08 2004 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified August 11 08:17 EDT 2022. Contains 356055 sequences. (Running on oeis4.)