Search: id:A014137
Results 1-1 of 1 results found.

%I A014137
%S A014137 1,2,4,9,23,65,197,626,2056,6918,23714,82500,290512,1033412,
%T A014137 3707852,13402697,48760367,178405157,656043857,2423307047,
%U A014137 8987427467,33453694487,124936258127,467995871777,1757900019101
%N A014137 Partial sums of Catalan numbers (A000108).
%C A014137 This is also the result of applying the transformation on generating 
               functions A -> 1/((1-x)*(1-x*A)) to the g.f. for the Catalan numbers.
%C A014137 p divides a(p)-3 for prime p=3 and p=7,13,19,31,37,43..=A002476 Primes 
               of form 6n + 1. p^2 divides a(p^2)-3 for prime p>3. - Alexander Adamchuk 
               (alex(AT)kolmogorov.com), Jul 11 2006
%C A014137 Prime p divides a(p) for p = {2, 3, 5, 11, 17, 23, 29, 41, 47, 53, 59, 
               71, 83, 89, 101, ...} = A045309 Primes congruent to {0, 2} mod 3; 
               and A045309 Primes p such that x^3 = n (integer) has only one solution 
               mod p. Nonprime numbers n such that n divides a(n) are listed in 
               A128287 = {1, 8, 133, ...}. - Alexander Adamchuk (alex(AT)kolmogorov.com), 
               Feb 23 2007
%C A014137 For p prime >=5, a(p-1) = 1 or -2 (mod p) according as p = 1 or -1 (mod 
               3) (see Pan and Sun link). For example, with p=5, a(p-1) = 23 = -2 
               (mod p). - David Callan (callan(AT)stat.wisc.edu), Nov 29 2007
%C A014137 Hankel transform is A010892(n+1). [From Paul Barry (pbarry(AT)wit.ie), 
               Apr 24 2009]
%C A014137 Equals INVERTi transform of A000245: (1, 3, 9, 28,...). [From Gary W. 
               Adamson (qntmpkt(AT)yahoo.com), May 15 2009]
%D A014137 N. S. S. Gu, N. Y. Li and T. Mansour, 2-Binary trees: bijections and 
               related issues, Discr. Math., 308 (2008), 1209-1221.
%D A014137 I. Pak, Partition identities and geometric bijections. Proc. Amer. Math. 
               Soc. 132 (2004), 3457-3462.
%H A014137 T. D. Noe, <a href="b014137.txt">Table of n, a(n) for n=0..200</a>
%H A014137 A. F. Labossiere, <a href="http://members.lycos.co.uk/sobalian/index.html">
               Sobalian Coefficients</a>.
%H A014137 A. F. Labossiere, <a href="http://members.lycos.co.uk/stereotomography/
               index.html">Miscellaneous</a>.
%H A014137 Hao Pan and Zhi-Wei Sun, <a href="http://front.math.ucdavis.edu/math.CO/
               0509648">A combinatorial identity with application to Catalan numbers</
               a>
%F A014137 G.f.: (1-(1-4*x)^(1/2))/(2*x*(1-x)).
%F A014137 Sum_{i=1..n} c(i) = Sum_{i=1..n} C(2*i-2, i-1)/i = 1/(n-1)! * [ n^(n-2) 
               +C(n, 2)*n^C(n-3, 1) +{8*C(n-4, 0) +19*C(n-4, 1) +24*C(n-4, 2) +14*C(n-4, 
               3) +3*C(n-4, 4)}*n^(n-4) +{18*C(n-5, 0) +82*C(n-5, 1) +229*C(n-5, 
               2) +323*C(n-5, 3) +244*C(n-5, 4) +95*C(n-5, 5) +15*C(n-5, 6)}*n^(n-5) 
               +... +C(n-3, 0)*(n-1)! ] (where c() = Catalan numbers A000108). - 
               Andre F. Labossiere (boronali(AT)laposte.net), May 17 2004
%F A014137 a(n) = Sum[(2k)!/(k!)^2/(k+1),{k,0,n}]. - Alexander Adamchuk (alex(AT)kolmogorov.com), 
               Jul 11 2006
%t A014137 Table[Sum[(2k)!/(k!)^2/(k+1),{k,0,n}],{n,1,30}] - Alexander Adamchuk 
               (alex(AT)kolmogorov.com), Jul 11 2006
%Y A014137 a(n) = A014138[n-1]+1.
%Y A014137 Cf. A000108, A094638, A001246, A033536, A000984, A094639, A006134, A082894, 
               A002897, A079727.
%Y A014137 Cf. A002476, A045309, A128287.
%Y A014137 A000245 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 15 2009]
%Y A014137 Sequence in context: A000083 A092668 A164039 this_sequence A007476 A129698 
               A117419
%Y A014137 Adjacent sequences: A014134 A014135 A014136 this_sequence A014138 A014139 
               A014140
%K A014137 nonn,nice
%O A014137 0,2
%A A014137 N. J. A. Sloane (njas(AT)research.att.com).

Search completed in 0.003 seconds