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A127632
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Expansion of c(x*c(x)), where c(x) is the g.f. for A000108.
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15
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1, 1, 3, 11, 44, 185, 804, 3579, 16229, 74690, 347984, 1638169, 7780876, 37245028, 179503340, 870374211, 4243141332, 20786340271, 102275718924, 505235129250, 2504876652190, 12459922302900, 62167152967680, 311040862133625
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OFFSET
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0,3
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COMMENTS
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Old name was: Expansion of 1/(1 - x*c(x) * c(x*c(x))), where c(x) is the g.f. of A000108.
Hankel transform appears to be A075845.
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
This is the number of intervals in the comb posets of Pallo. See the Pallo and Csar et al. references for the definition of these posets. For the proof, see the Aval et al. reference - 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
a(n) is the number of uniquely sorted permutations of length 2n+1 that avoid the patterns 231 and 4132. (A permutation is called uniquely sorted if it has exactly one preimage under West's stack-sorting map. See the Defant link.) - Colin Defant, Jun 08 2019
a(n) is the number of 132-avoiding permutations of length 3*n whose disjoint cycle decomposition contains only 3-cycles (a,b,c) with a>b>c. See the Archer and Graves reference. - Alexander Burstein, Oct 21 2021
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LINKS
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FORMULA
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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_{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
G.f.: A(x) = 1 + x*c(x)^2*m(x*c(x)^2), where m(x) is the g.f. of A001006 and c(x) is the g.f. of A000108.
G.f.: A(x) satisfies: A(-x*A(x)^5) = 1/A(x). (End)
a(n) = Catalan(n - 1) * hypergeom([3/2, 2, 1 - n], [3, 2 - 2*n], 4) for n >= 1.
The G.f. satisfies the algebraic equation 0 = F^4*x - F^3 + 2*F^2 - 2*F + 1. - F. Chapoton, Oct 18 2021
D-finite with recurrence 3*n*(n-1)*(n+1)*a(n) -4*n*(7*n-2)*(n-1)*a(n-1) +8*(n-1)*(2*n^2+30*n-65)*a(n-2) +8*(56*n^3-520*n^2+1534*n-1445)*a(n-3) -32*(4*n-15)*(2*n-7)*(4*n-13)*a(n-4)=0. - R. J. Mathar, Aug 01 2022
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MAPLE
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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:
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MATHEMATICA
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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 *)
a[n_] := CatalanNumber[n - 1] HypergeometricPFQ[{3/2, 2, 1 - n}, {3, 2 - 2 n}, 4];
a[0] := 1; Table[a[n], {n, 0, 23}] (* Peter Luschny, May 12 2021 *)
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PROG
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(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
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CROSSREFS
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Row sums of number triangle A127631.
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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