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 A125177 Triangle read by rows: T(n,0)=C(2n,n)/(n+1) for n>=0; T(0,k)=0 for k>=1; T(n,k)=T(n-1,k)+T(n-1,k-1) for n>=1, k>=1. 4
 1, 1, 1, 2, 2, 1, 5, 4, 3, 1, 14, 9, 7, 4, 1, 42, 23, 16, 11, 5, 1, 132, 65, 39, 27, 16, 6, 1, 429, 197, 104, 66, 43, 22, 7, 1, 1430, 626, 301, 170, 109, 65, 29, 8, 1, 4862, 2056, 927, 471, 279, 174, 94, 37, 9, 1, 16796, 6918, 2983, 1398, 750, 453, 268, 131, 46, 10, 1, 58786 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Column k (k>=1) starts with 0, followed by the partial sums of column k-1. Row sums yield A126221. Indexing n and k from 1 instead of from 0, T(n,k) is the number of Dyck n-paths whose first peak is at height k and whose first component avoids DUU. A primitive Dyck path is one whose only return (to ground level) is at the end. The interior returns of a general Dyck path split the path into a list of primitive Dyck paths, called its components. For example, UUDDUD has components UUDD, UD and T(4,2) = 4 counts UUDUDUDD, UUDDUUDD, UUDDUDUD, UUDUDDUD (but not UUDUUDDD because its first component contains a DUU). - David Callan, Jan 17 2007 Riordan array (c(x),x/(1-x)), c(x) the g.f. of A000108. Equal to ((1-x)*c(x),x)*A007318. [Paul Barry, May 06 2009] LINKS Seiichi Manyama, Table of n, a(n) for n = 0..10000 FORMULA G.f.: G(t,x)=(1-x)[1-sqrt(1-4x)]/[2x(1-x-tx)]. T(n,k) = Sum_{j=0..n} C(n-j,k)*if(j=0,0^j, A000108(j)-A000108(j-1)). [Paul Barry, May 06 2009] T(n,k) = Sum_{i=0..n-k} binomial(n-i-1,n-k-i)*A000108(i). - Vladimir Kruchinin, Nov 03 2016 EXAMPLE First few rows of the triangle are: 1; 1, 1; 2, 2, 1; 5, 4, 3, 1; 14, 9, 7, 4, 1; 42, 23, 16, 11, 5, 1; ... (5,3) = 16 = 7 + 9 = (4,3) + (4,2). Contribution from Paul Barry, May 06 2009: (Start) Production matrix is 1, 1, 1, 1, 1, 1, 0, 1, 1, 2, 0, 0, 1, 1, 4, 0, 0, 0, 1, 1, 9, 0, 0, 0, 0, 1, 1, 21, 0, 0, 0, 0, 0, 1, 1, 51, 0, 0, 0, 0, 0, 0, 1, 1, 127, 0, 0, 0, 0, 0, 0, 0, 1, 1 (End) MAPLE T:=proc(n, k) if k=0 then binomial(2*n, n)/(n+1) elif n=0 then 0 else T(n-1, k)+T(n-1, k-1) fi end: for n from 0 to 11 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form G:=(1-x)*(1-sqrt(1-4*x))/2/x/(1-x-t*x): Gser:=simplify(series(G, x=0, 15)): for n from 0 to 11 do P[n]:=sort(coeff(Gser, x, n)) od: for n from 0 to 11 do seq(coeff(P[n], t, j), j=0..n) od; # yields sequence in triangular form PROG (Maaxima) T(n, k)=sum((binomial(2*i, i)*binomial(n-i-1, n-k-i))/(i+1), i, 0, n-k); /* Vladimir Kruchinin, Nov 03 2016 */ CROSSREFS Cf. A000108, A125178, A126221. Sequence in context: A105292 A273342 A276067 * A125178 A101975 A136388 Adjacent sequences:  A125174 A125175 A125176 * A125178 A125179 A125180 KEYWORD nonn,tabl AUTHOR Gary W. Adamson, Nov 22 2006 EXTENSIONS Edited by Emeric Deutsch, Dec 28 2006 STATUS approved

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Last modified October 17 22:05 EDT 2019. Contains 328134 sequences. (Running on oeis4.)