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%I
%S 1,3,11,42,163,638,2510,9908,39203,155382,616666,2449868,9740686,
%T 38754732,154276028,614429672,2448023843,9756737702,38897306018,
%U 155111585372,618679078298,2468152192772
%N Sum(binomial(2*n,i),i=0..n).
%C 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_, Apr 01 2002
%C 1 1 1 1 1 1 1 ...
%C 2 3 4 5 6 7 8 ...
%C 4 7 11 16 22 ....
%C 8 15 26 42 64....
%C 16 31 ..99 163...
%C Hankel transform is n+1. - _Paul Barry_, Jan 11 2007
%C Contribution from _Gary W. Adamson_, Dec 27 2008: (Start)
%C A032443 is an analogue of the Catalan sequence: Let M = an infinite Cartan matrix (-1's in the super and sub-diagonals and (2,2,2,...) in the main diagonal which we modify to (1,2,2,2,...). Then A000108 can be generated by 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 Convolved with the Catalan sequence, A000108: (1, 1, 2, 5, 14,...) = powers of 4, A000302: (1, 4, 16, 64,...). [From _Gary W. Adamson_, May 15 2009]
%C Row sums of A094527. [From _Paul Barry_, Sep 07 2009]
%C Hankel transform of the aeration of this sequence is C(n+2,2). [From _Paul Barry_, Sep 26 2009]
%C Number of 4-ary words of length n in which the number of 1's does not exceed the number of 0's. - _David Scambler_, Aug 14 2012
%D A. Bernini, F. Disanto, R. Pinzani and S. Rinaldi, Permutations defining convex permutominoes, preprint, 2007.
%D M. Klazar, Twelve countings with rooted plane trees, European Journal of Combinatorics 18 (1997), 195-210; Addendum, 18 (1997), 739-740.
%D D. Phulara and L. W. Shapiro, Descendants in ordered trees with a marked vertex, Congressus Numerantium, 205 (2011), 121-128.
%H Mircea Merca, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Merca1/merca6.html">A Note on Cosine Power Sums</a> J. Integer Sequences, Vol. 15 (2012), Article 12.5.3.
%F a(n) = (4^n+binomial(2*n, n))/2 (David W. Wilson)
%F a(n)=sum_{0<=i_1<=i_2<=n} binomial(n, i_2)*binomial(n, i_1+i_2) - _Benoit Cloitre_, Oct 14 2004
%F 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_, Jan 14 2005
%F a(n) = sum{k=0..n, C(n+k-1,k)*2^(n-k)}; - _Paul Barry_, Sep 28 2007
%F E.g.f.: exp(2*x)*(exp(2*x) + BesselI(0,2*x))/2. For BesselI see Abramowitz-Stegun (reference and link under A008277), p. 375, eq. 9.6.10. See also A000984 for its e.g.f. given by M. Somos, S. N. Gladkovskii, and a W. Lang comment. - Wolfdieter Lang, Jan 16 2012
%F From Sergei N. Gladkovskii, Aug 13 2012: (Start)
%F G.f. (1/sqrt(1-4*x) + 1/(1-4*x))/2 = G(0)/2 where G(k)= 1 + ((2*k)!)/(k!)^2/(4^k - 4*x*(16^k)/( 4*x*(4^k) + (2*k)!)/(k!)^2/G(k+1))); (continued fraction, 3rd kind, 3-step).
%F E.g.f. G(0)/2 where G(k)= 1 + ((2*k)!)/(k!)^2/(4^k - 4*x*(16^k)/( 4*x*(4^k) + (k+1)*(2*k)!)/(k!)^2/G(k+1))); (continued fraction, 3rd kind, 3-step).
%F (End)
%F O.g.f.: (1 - x*(2 + c(x)))/(1 - 4*x)^(3/2), with c the o.g.f. of A000108 (Catalan). - _Wolfdieter Lang_, Nov 22 2012
%F n*a(n) +2*(-4*n+3)*a(n-1) +8*(2*n-3)*a(n-2)=0. - _R. J. Mathar_, Dec 04 2012
%Y Binomial transform of A027914. Hankel transform is {1, 2, 3, 4, ..., n, ...} - _John W. Layman_, Aug 04 2000
%Y A000108 [From _Gary W. Adamson_, Dec 27 2008]
%Y A000302 [From _Gary W. Adamson_, May 15 2009]
%K nonn
%O 0,2
%A J. H. Conway (conway(AT)math.princeton.edu)
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