OFFSET
0,3
COMMENTS
a(n) = p(n + 1) where p(x) is the unique degree-n polynomial such that p(k) = Catalan(k) for k = 0, 1, ..., n. - Michael Somos, Jan 05 2012
Number of Dyck (n+1)-paths whose minimum ascent length is 1. - David Scambler, Aug 22 2012
From Alois P. Heinz, Jun 29 2014: (Start)
a(n) is the number of ordered rooted trees with n+2 nodes such that the minimal outdegree equals 1. a(2) = 4:
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(End)
Number of non-crossing partitions of {1,2,..,n+1} that contain cyclical adjacencies. a(2) = 4, [12|3, 13|2, 1|23, 123]. - Yuchun Ji, Nov 13 2020
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: (sqrt( 1 - 2*x - 3*x^2 ) / (1 + x) - sqrt( 1 - 4*x )) / (2*x^2) = 2 / (sqrt( 1 - 2*x - 3*x^2 ) + (1 + x) * sqrt( 1 - 4*x )). - Michael Somos, Jan 05 2012
a(n) ~ 2^(2*n+2) / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jan 21 2017
EXAMPLE
1 + x + 4*x^2 + 11*x^3 + 36*x^4 + 117*x^5 + 393*x^6 + 1339*x^7 + 4630*x^8 + ...
a(2) = 4 since p(x) = (x^2 - x + 2) / 2 interpolates p(0) = 1, p(1) = 1, p(2) = 2, and p(3) = 4. - Michael Somos, Jan 05 2012
MAPLE
a:= proc(n) option remember; `if`(n<3, [1, 1, 4][n+1],
((30*n^3-44*n^2-22*n+24)*a(n-1)-(25*n^3-105*n^2+140*n-48)*a(n-2)
-6*(n-1)*(5*n-4)*(2*n-3)*a(n-3))/(n*(n+2)*(5*n-9)))
end:
seq(a(n), n=0..30); # Alois P. Heinz, Jun 29 2014
MATHEMATICA
max = 30; t = Table[Differences[Table[CatalanNumber[k], {k, 0, max}], n], {n, 0, max}]; a[n_] := Sum[t[[n-k+1, k]], {k, 1, n}]; Array[a, max] (* Jean-François Alcover, Jan 21 2017 *)
PROG
(PARI) {a(n) = if( n<0, 0, n++; subst( polinterpolate( vector(n, k, binomial( 2*k - 2, k - 1) / k)), x, n + 1))} /* Michael Somos, Jan 05 2012 */
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( 2 / (sqrt( 1 - 2*x - 3*x^2 + A) + (1 + x) * sqrt( 1 - 4*x + A)) , n))} /* Michael Somos, Jan 05 2012 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Philippe Deléham, May 26 2005
EXTENSIONS
Typo in a(20) corrected and more terms from Alois P. Heinz, Jun 29 2014
STATUS
approved