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 A068763 Array of numbers related to certain generalized Catalan sequences. 9
 1, 1, 1, 2, 2, 5, 6, 1, 14, 20, 6, 42, 70, 30, 2, 132, 252, 140, 20, 429, 924, 630, 140, 5, 1430, 3432, 2772, 840, 70, 4862, 12870, 12012, 4620, 630, 14, 16796, 48620, 51480, 24024, 4620, 252, 58786, 184756, 218790 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS The row length sequence of this array is [1,2,2,3,3,4,4,5,5,...] = A008619(n+2), n>=0. The row polynomials p(n,x) := Sum_{m=0..floor((n+1)/2)} a(n,m)*x^m produce, for x = (b-a^2)/a^2 (not 0), the two parameter family of sequences K(a,b; n) := (a^(n+1))*p(n,(b-a^2)/a^2) with g.f. K(a,b; x) := (1-sqrt(1-4*x*(a+x*(b-a^2))))/(2*x). Some members are: K(1,1; n)=A000108(n) (Catalan), K(1,2; n)=A025227(n-1), K(2,1; n)=A025228(n-1), K(1,3; n)=A025229(n-1), K(3,1; n)=A025230(n-1). For a=b=2..10 the sequences K(a,a; n)/a are A068764-A068772. The column sequences (without leading 0's) are: A000108 (Catalan), A000984 (central binomial), A002457, 2*A002802, 5*A020918, 14*A020920, 42*A020922, ... a(n,m) is the number of ways to designate exactly m cherries over all binary trees with n internal nodes.  A cherry is an internal node whose descendants are both external nodes.  Cf. A091894 which gives the number of binary trees with m cherries. - Geoffrey Critzer, Jul 24 2020 LINKS P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 213. FORMULA a(n, m) = binomial(n+1-m, m)*C(n-m) if 0<= m <= floor((n+1)/2), else 0, with C(n) := A000108(n) (Catalan). G.f. for column m=1, 2, ...: (x^(2*m-1))*C(m-1)/(1-4*x)^((2*m-1)/2); m=0: c(x), g.f. for A000108 (Catalan). G.f. for row polynomials p(n, x): c(z)+x*z*c(x*(z^2)/(1-4*z))/sqrt(1-4*z) = (1-sqrt(1-4*z*(1+x*z)))/(2*z), where c(x) is the g.f. of A000108 (Catalan). G.f. for triangle: (1 - sqrt(1 - 4*x (1 + y*x)))/(2*x). - Geoffrey Critzer, Jul 24 2020 EXAMPLE {1}; {1,1}; {2,2}; {5,6,1}; {14,20,6}; ...; p(3,x) = 5+6*x+x^2. MATHEMATICA nn = 10; b[z_] := (1 - Sqrt[1 - 4 z])/(2 z); Map[Select[#, # > 0 &] &, CoefficientList[Series[v b[v z] /. v -> (1 + u z ), {z, 0, nn}], {z, u}]] // Grid (* Geoffrey Critzer, Jul 24 2020 *) CROSSREFS Cf. A025227(n-1) (row sums). Sequence in context: A063501 A103892 A000403 * A176989 A250303 A301477 Adjacent sequences:  A068760 A068761 A068762 * A068764 A068765 A068766 KEYWORD nonn,easy,tabf AUTHOR Wolfdieter Lang, Mar 04 2002 STATUS approved

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Last modified October 1 04:06 EDT 2020. Contains 337441 sequences. (Running on oeis4.)