Search: id:A032443
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%I A032443
%S A032443 1,3,11,42,163,638,2510,9908,39203,155382,616666,2449868,
%T A032443 9740686,38754732,154276028,614429672,2448023843,9756737702,
%U A032443 38897306018,155111585372,618679078298,2468152192772
%N A032443 Sum(binomial(2*n,i),i=0..n).
%C A032443 Array interpretation : first row is filled with 1's, first column with 
               powers of 2, b(i,j)=b(i-1,j)+b(i,j-1); then a(n)=b(n,n) - Benoit 
               Cloitre (benoit7848c(AT)orange.fr), Apr 01 2002
%C A032443 1 1 1 1 1 1 1 ...
%C A032443 2 3 4 5 6 7 8 ...
%C A032443 4 7 11 16 22 ....
%C A032443 8 15 26 42 64....
%C A032443 16 31 ..99 163...
%C A032443 Hankel transform is n+1. - Paul Barry (pbarry(AT)wit.ie), Jan 11 2007
%C A032443 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 27 2008: 
               (Start)
%C A032443 A032443 is an analogue of the Catalan sequence: Let M = an infinite Cartan
%C A032443 matrix (-1's in the super and sub-diagonals and (2,2,2,...) in the main
%C A032443 diagonal which we modify to (1,2,2,2,...). Then A000108 can be generated 
               by
%C A032443 accessing the leftmost term in M^n * [1,0,0,0,...]. Change the operation 
               to M^n * [1,2,3,...] to get A032443. Or, take iterates M * [1,2,3,
               ...] -> M * ANS, -> M * ANS,...; accessing the leftmost term. (End)
%C A032443 Convolved with the Catalan sequence, A000108: (1, 1, 2, 5, 14,...) = 
               powers of 4, A000302: (1, 4, 16, 64,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               May 15 2009]
%C A032443 Row sums of A094527. [From Paul Barry (pbarry(AT)wit.ie), Sep 07 2009]
%C A032443 Hankel tranform of the aeration of this sequence is C(n+2,2). [From Paul 
               Barry (pbarry(AT)wit.ie), Sep 26 2009]
%D A032443 A. Bernini, F. Disanto, R. Pinzani and S. Rinaldi, Permutations defining 
               convex permutominoes, preprint, 2007.
%D A032443 M. Klazar, Twelve countings with rooted plane trees, European Journal 
               of Combinatorics 18 (1997), 195-210; Addendum, 18 (1997), 739-740.
%F A032443 (4^n+binomial(2*n, n))/2 (David W. Wilson)
%F A032443 a(n)=sum_{0<=i_1<=i_2<=n}binomial(n, i_2)*binomial(n, i_1+i_2) - Benoit 
               Cloitre (benoit7848c(AT)orange.fr), Oct 14 2004
%F A032443 Sequence with interpolated zeros has a(n)=sum{k=0..floor(n/2), if(mod(n-2k, 
               2)=0, C(n, k), 0)}. - Paul Barry (pbarry(AT)wit.ie), Jan 14 2005
%F A032443 a(n)=sum{k=0..n, C(n+k-1,k)2^(n-k)}; - Paul Barry (pbarry(AT)wit.ie), 
               Sep 28 2007
%Y A032443 Binomial transform of A027914. Hankel transform is {1, 2, 3, 4, ..., 
               n, ...} - John W. Layman (layman(AT)math.vt.edu), Aug 04 2000
%Y A032443 A000108 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 27 2008]
%Y A032443 A000302 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 15 2009]
%Y A032443 Sequence in context: A106460 A059716 A122368 this_sequence A143464 A117641 
               A084782
%Y A032443 Adjacent sequences: A032440 A032441 A032442 this_sequence A032444 A032445 
               A032446
%K A032443 nonn
%O A032443 0,2
%A A032443 J. H. Conway (conway(AT)math.princeton.edu)

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