Search: id:A032357
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%I A032357
%S A032357 1,0,2,3,11,31,101,328,1102,3760,13036,45750,162262,580638,2093802,
%T A032357 7601043,27756627,101888163,375750537,1391512653,5172607767,19293659253,
%U A032357 72188904387,270870709263,1019033438061,3842912963391,14524440108761
%N A032357 Convolution of Catalan numbers and powers of -1.
%C A032357 Absolute value of the alternating sum of Catalan Numbers. - Alexander 
               Adamchuk (alex(AT)kolmogorov.com), Jul 03 2006
%C A032357 Sums of two consecutive terms are A032357[n-1]+A032357[n]=1,2,5,14,42..=A000108[n] 
               Catalan Numbers. The prime p divides a((p-3)/2) for p=11,19,29,31,
               41,59,61,71..=A045468[n] Primes congruent to {1, 4} mod 5. Prime 
               p divides a(2p+1) for p=5,11,19,29,31,41,59,61,71..=A038872[n] Primes 
               congruent to {0, 1, 4} mod 5. Also odd primes where 5 is a square 
               mod p. - Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 03 2006
%C A032357 Hankel transform is F(2n+1). - Paul Barry (pbarry(AT)wit.ie), Jul 22 
               2008
%C A032357 Equals INVERTi transform of A000958 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Apr 10 2009]
%C A032357 Inverse binomial transform of A002212 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Sep 17 2009]
%F A032357 G.f.: c(x)/(1+x), where c(x) = g.f. for Catalan numbers.
%F A032357 Sum((-1)^(n-k)*C(k), k=0..n), C(k)=A00108(k), Catalan numbers
%F A032357 a(n)=((-1)^(n+1)-binomial(2*(n+1), n+1)*sum((-5)^k*binomial(n+1, k)/binomial(2*k, 
               k), k=0..n+1))/2
%F A032357 Also: a(n)=C[2n, n]/(n+1)-a(n-1)=A000108[n]-a(n-1) with a(0)=1. - Labos 
               E. (labos(AT)ana.hu), Apr 26 2003
%F A032357 a(n) = Sum[ (-1)^(k+n)*CatalanNumber[k], {k,0,n}]. - Alexander Adamchuk 
               (alex(AT)kolmogorov.com), Jul 03 2006
%t A032357 Table[Sum[(-1)^(k+n)*CatalanNumber[k],{k,0,n}],{n,0,60}] - Alexander 
               Adamchuk (alex(AT)kolmogorov.com), Jul 03 2006
%Y A032357 Cf. A000108.
%Y A032357 Cf. A000108, A014137, A014138, A033297, A045468, A038872, A064739.
%Y A032357 A000958 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 10 2009]
%Y A032357 Sequence in context: A038987 A142957 A080155 this_sequence A144056 A062630 
               A159458
%Y A032357 Adjacent sequences: A032354 A032355 A032356 this_sequence A032358 A032359 
               A032360
%K A032357 easy,nonn
%O A032357 0,3
%A A032357 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
%E A032357 More terms from Christian G. Bower (bowerc(AT)usa.net), Apr 15 1998.
%E A032357 More terms from Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 03 2006

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