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A106640 Row sums of A059346.

%I A106640 #45 Dec 18 2020 14:25:56
%S A106640 1,1,4,11,36,117,393,1339,4630,16193,57201,203799,731602,2643903,
%T A106640 9611748,35130195,129018798,475907913,1762457595,6550726731,
%U A106640 24428808690,91377474411,342763939656,1289070060903,4859587760076,18360668311027,69514565858653,263693929034909
%N A106640 Row sums of A059346.
%C A106640 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
%C A106640 Number of Dyck (n+1)-paths whose minimum ascent length is 1. - _David Scambler_, Aug 22 2012
%C A106640 From _Alois P. Heinz_, Jun 29 2014: (Start)
%C A106640 a(n) is the number of ordered rooted trees with n+2 nodes such that the minimal outdegree equals 1. a(2) = 4:
%C A106640    o    o      o      o
%C A106640    |    |     / \    / \
%C A106640    o    o    o   o  o   o
%C A106640    |   / \   |          |
%C A106640    o  o   o  o          o
%C A106640    |
%C A106640    o
%C A106640 (End)
%C A106640 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
%H A106640 Alois P. Heinz, <a href="/A106640/b106640.txt">Table of n, a(n) for n = 0..1000</a>
%F A106640 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
%F A106640 a(n) = A000108(n+1) - A005043(n+1).
%F A106640 a(n) ~ 2^(2*n+2) / (sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jan 21 2017
%F A106640 a(n) = A000296(n+2) - A247494(n+1); i.e., remove the crossing partitions from the partitions with cyclical adjacencies. - _Yuchun Ji_, Nov 17 2020
%e A106640 1 + x + 4*x^2 + 11*x^3 + 36*x^4 + 117*x^5 + 393*x^6 + 1339*x^7 + 4630*x^8 + ...
%e A106640 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
%p A106640 a:= proc(n) option remember; `if`(n<3, [1, 1, 4][n+1],
%p A106640       ((30*n^3-44*n^2-22*n+24)*a(n-1)-(25*n^3-105*n^2+140*n-48)*a(n-2)
%p A106640        -6*(n-1)*(5*n-4)*(2*n-3)*a(n-3))/(n*(n+2)*(5*n-9)))
%p A106640     end:
%p A106640 seq(a(n), n=0..30);  # _Alois P. Heinz_, Jun 29 2014
%t A106640 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 *)
%o A106640 (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 */
%o A106640 (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 */
%Y A106640 Cf. A000108, A005043, A244455, A000296, A247494.
%K A106640 nonn
%O A106640 0,3
%A A106640 _Philippe Deléham_, May 26 2005
%E A106640 Typo in a(20) corrected and more terms from _Alois P. Heinz_, Jun 29 2014