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A127632 Expansion of c(x*c(x)), where c(x) is the g.f. for A000108. 9
1, 1, 3, 11, 44, 185, 804, 3579, 16229, 74690, 347984, 1638169, 7780876, 37245028, 179503340, 870374211, 4243141332, 20786340271, 102275718924, 505235129250, 2504876652190, 12459922302900, 62167152967680, 311040862133625 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Old name was: Expansion of 1/(1 - x*c(x) * c(x*c(x))), where c(x) is the g.f. of A000108.

Row sums of number triangle A127631.

Hankel transform appears to be A075845.

Catalan transform of Catalan numbers. - Philippe Deléham, Jun 20 2007

Number of functions f:[1,n] -> [1,n] satisfying the condition that, for all i<j, f(j)-(j-i) is not in the interval [1,f(i)-1]; see the Callan reference. - Joerg Arndt, May 31 2013

Seems to give (checked up to a(10)=347984) the number of intervals in the comb posets of Pallo, see the Pallo and Csar et al. references for the definition of these posets. - F. Chapoton, Apr 06 2015

Construct a lower triangular array (T(n,k))n,k>=0 by putting the sequence of Catalan numbers as the first column of the array and completing the remaining columns using the recurrence T(n,k) = T(n,k-1) + T(n-1,k). This sequence will then be the leading diagonal of the array. - Peter Bala, May 13 2017

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

P. Bala, A note on the Catalan transform of a sequence

David Callan, A combinatorial interpretation of the Catalan transform of the Catalan numbers, arXiv:1111.0996 [math.CO], 2011.

David Callan, Permutations avoiding 4321 and 3241 have an algebraic generating function, arXiv:1306.3193 [math.CO], 2013.

S. Csar, R. Sengupta, W. Suksompong, On a Subposet of the Tamari Lattice, arXiv preprint arXiv:1108.5690 [math.CO], 2011.

Tian-Xiao He, Louis W. Shapiro, Row sums and alternating sums of Riordan arrays, Linear Algebra and its Applications, Volume 507, 15 October 2016, Pages 77-95.

J. M. Pallo, Right-arm rotation distance between binary trees, Inform. Process. Lett., 87(4):173-177, 2003.

Y. Sun, Z. Wang, Consecutive pattern avoidances in non-crossing trees, Graph. Combinat. 26 (2010) 815-832, table 1, {ud}

S. Yakoubov, Pattern Avoidance in Extensions of Comb-Like Posets, arXiv preprint arXiv:1310.2979 [math.CO], 2013.

FORMULA

a(n) = A127714(n+1,2n+1).

G.f. A(x) satisfies: 0 = 1 - A(x) + A(x)^2 * x * c(x) where c(x) is the g.f. of A000108.

G.f.: 2/(1 + sqrt( 2 * sqrt(1 -4*x) - 1)). - Michael Somos, May 04 2007

a(n) = Sum_{k=0..n} A106566(n,k)*A000108(k). - Philippe Deléham, Jun 20 2007

a(n) = (Sum_{m=1..n} (m*Sum_{k=m..n} binomial(2*k-m-1,k-1)*binomial(2*n-k-1,n-1)))/n, a(0)=1. - Vladimir Kruchinin, Oct 08 2011

Conjecture: 3*n*(n-1)*(4*n-9)*(n+1)*a(n) - 8*n*(n-1)*(20*n^2-67*n+48)*a(n-1) + 8*(4*n-5)*(n-1)*(22*n^2-94*n+99)*a(n-2) - 8*(4*n-11)*(4*n-5)*(4*n-9)*(2*n-5)*a(n-3) = 0. - R. J. Mathar, May 04 2018

a(n) ~ 2^(4*n - 1/2) / (sqrt(Pi) * n^(3/2) * 3^(n - 1/2)). - Vaclav Kotesovec, Aug 14 2018

MAPLE

a:= proc(n) option remember; `if`(n<3, [1, 1, 3][n+1],

      ((8*(4*n-11))*(4*n-5)*(4*n-9)*(2*n-5)*a(n-3)

      -(8*(4*n-5))*(n-1)*(22*n^2-94*n+99)*a(n-2)

      +8*n*(n-1)*(20*n^2-67*n+48)*a(n-1))/

      ((3*(4*n-9))*(n+1)*n*(n-1)))

    end:

seq(a(n), n=0..30);  # Alois P. Heinz, Apr 06 2015

MATHEMATICA

a[n_] := Sum[m*(2*n-m-1)!*HypergeometricPFQ[{m/2+1/2, m/2, m-n}, {m, m-2*n+1}, 4]/(n!*(n-m)!), {m, 1, n}]; a[0]=1; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jul 24 2012, after Vladimir Kruchinin *)

PROG

(PARI) {a(n)= if(n<1, n==0, polcoeff( serreverse( x*(1-x)^3*(1-x^3)/(1-x^2)^4 +x*O(x^n) ), n))} /* Michael Somos, May 04 2007 */

(PARI) {a(n)= local(A); if(n<1, n==0, A= serreverse( x-x^2 +x*O(x^n) ); polcoeff( 1/(1 - subst(A, x, A)), n))} /* Michael Somos, May 04 2007 */

(Maxima) a(n):=if n=0 then 1 else sum(m*sum(binomial(2*k-m-1, k-1)*binomial(2*n-k-1, n-1), k, m, n), m, 1, n)/n; \\ Vladimir Kruchinin, Oct 08 2011

CROSSREFS

Cf. A127714, A075845, A000108, A009766.

Sequence in context: A091200 A271931 A151105 * A061706 A167012 A167013

Adjacent sequences:  A127629 A127630 A127631 * A127633 A127634 A127635

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Jan 20 2007, Jan 25 2007

EXTENSIONS

Better name from David Callan, Jun 03 2013

STATUS

approved

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Last modified October 15 22:50 EDT 2018. Contains 316252 sequences. (Running on oeis4.)