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 A127632 Expansion of c(x*c(x)), where c(x) is the g.f. for A000108. 9

%I

%S 1,1,3,11,44,185,804,3579,16229,74690,347984,1638169,7780876,37245028,

%T 179503340,870374211,4243141332,20786340271,102275718924,505235129250,

%U 2504876652190,12459922302900,62167152967680,311040862133625

%N Expansion of c(x*c(x)), where c(x) is the g.f. for A000108.

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

%C Row sums of number triangle A127631.

%C Hankel transform appears to be A075845.

%C Catalan transform of Catalan numbers. - _Philippe Deléham_, Jun 20 2007

%C 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

%C Seems to give (checked up to a(10)=347984) the number of intervals in the comb posets of Pallo, see the Pallo and Csar & alii references for the definition of these posets. - _F. Chapoton_, Apr 06 2015

%C 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

%H Alois P. Heinz, <a href="/A127632/b127632.txt">Table of n, a(n) for n = 0..1000</a>

%H P. Bala, <a href="/A001517/a001517.pdf">A note on the Catalan transform of a sequence</a>

%H David Callan, <a href="http://arxiv.org/abs/1111.0996">A combinatorial interpretation of the Catalan transform of the Catalan numbers</a>, arXiv:1111.0996 [math.CO], 2011.

%H David Callan, <a href="https://arxiv.org/abs/1306.3193">Permutations avoiding 4321 and 3241 have an algebraic generating function</a>, arXiv:1306.3193 [math.CO], 2013.

%H S. Csar, R. Sengupta, W. Suksompong, <a href="http://arxiv.org/abs/1108.5690">On a Subposet of the Tamari Lattice</a>, arXiv preprint arXiv:1108.5690 [math.CO], 2011.

%H Tian-Xiao He, Louis W. Shapiro, <a href="https://doi.org/10.1016/j.laa.2016.05.035">Row sums and alternating sums of Riordan arrays</a>, Linear Algebra and its Applications, Volume 507, 15 October 2016, Pages 77-95.

%H J. M. Pallo, <a href="http://dx.doi.org/10.1016/S0020-0190(03)00283-7">Right-arm rotation distance between binary trees</a>, Inform. Process. Lett., 87(4):173-177, 2003.

%H Y. Sun, Z. Wang, <a href="http://dx.doi.org/10.1007/s00373-010-0950-9">Consecutive pattern avoidances in non-crossing trees</a>, Graph. Combinat. 26 (2010) 815-832, table 1, {ud}

%H S. Yakoubov, <a href="http://arxiv.org/abs/1310.2979">Pattern Avoidance in Extensions of Comb-Like Posets</a>, arXiv preprint arXiv:1310.2979 [math.CO], 2013.

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

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

%F G.f.: 2/(1 + sqrt( 2 * sqrt(1 -4*x) - 1)). - _Michael Somos_, May 04 2007

%F a(n) = Sum_{k, 0<=k<=n}A106566(n,k)*A000108(k). - _Philippe Deléham_, Jun 20 2007

%F 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

%F 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

%F a(n) ~ 2^(4*n - 1/2) / (sqrt(Pi) * n^(3/2) * 3^(n - 1/2)). - _Vaclav Kotesovec_, Aug 14 2018

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

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

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

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

%p ((3*(4*n-9))*(n+1)*n*(n-1)))

%p end:

%p seq(a(n), n=0..30); # _Alois P. Heinz_, Apr 06 2015

%t 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_ *)

%o (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 */

%o (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 */

%o (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

%Y Cf. A127714, A075845, A000108, A009766.

%K easy,nonn,changed

%O 0,3

%A _Paul Barry_, Jan 20 2007, Jan 25 2007

%E Better name from _David Callan_, Jun 03 2013

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Last modified August 21 14:53 EDT 2018. Contains 313954 sequences. (Running on oeis4.)