Search: id:A003168
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%I A003168 M3574
%S A003168 1,1,4,21,126,818,5594,39693,289510,2157150,16348960,125642146,
%T A003168 976789620,7668465964,60708178054,484093913917,3884724864390,
%U A003168 31348290348086,254225828706248,2070856216759478,16936016649259364
%N A003168 Number of blobs with 2n+1 edges.
%C A003168 a(n)= # of ways to dissect a convex (2n+2)-gon with non-crossing diagonals 
               so that no (2m+1)-gons (m>0) appear - Len Smiley (smiley(AT)math.uaa.alaska.edu)
%D A003168 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A003168 L. Carlitz, Enumeration of two-line arrays, Fib. Quart., 11 (1973), 113-130.
%D A003168 R. C. Read, On the enumeration of a class of plane multigraphs, Aequat. 
               Math., 31 (1986), 47-63.
%H A003168 T. D. Noe, <a href="b003168.txt">Table of n, a(n) for n=0..100</a>
%H A003168 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=415">
               Encyclopedia of Combinatorial Structures 415</a>
%H A003168 L. Smiley, <a href="http://www.math.uaa.alaska.edu/~smiley/vsd2.html">
               Even-gon reference</a>
%F A003168 Sum_{k=1..n} binomial(n, k)*binomial(2n+k, k-1)/n.
%F A003168 A(x)=Sum_{n>=0}a(n)x^(2n+1) satisfies (A-2*A^3)/(1-A^2)=x - Len Smiley 
               (smiley(AT)math.uaa.alaska.edu).
%F A003168 4*n*(2*n + 1)*(17*n - 22)*a(n) = (1207*n^3 - 2769*n^2 + 1850*n - 360)*a(n 
               - 1) - 2*(17*n - 5)*(n - 2)*(2*n - 3)*a(n - 2). - Vladeta Jovovic 
               (vladeta(AT)eunet.rs), Jul 16 2004
%F A003168 G.f.: A(x) = 1/(1-G003169(x)) where G003169(x) is the g.f. of A003169. 
               - Paul D. Hanna (pauldhanna(AT)juno.com), Nov 17 2004
%e A003168 a(2)=4 because we may place exactly one diagonal in 3 ways (forming 2 
               quadrilaterals), or not place any (leaving 1 hexagon).
%p A003168 Order := 40; solve(series((A-2*A^3)/(1-A^2),A)=x,A);
%o A003168 (PARI) a(n)=if(n<0,0,polcoeff(serreverse((x-2*x^3)/(1-x^2)+O(x^(2*n+2))),
               2*n+1))
%o A003168 (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=(1+x*A)/(1-x*A)^2); sum(k=0,
               n,polcoeff(A^(n-k),k))} (Hanna)
%Y A003168 Cf. A049124 (no 2m-gons).
%Y A003168 Cf. A003169, A100327.
%Y A003168 Sequence in context: A153291 A093965 A162480 this_sequence A032326 A007345 
               A099250
%Y A003168 Adjacent sequences: A003165 A003166 A003167 this_sequence A003169 A003170 
               A003171
%K A003168 nonn,easy,nice
%O A003168 0,3
%A A003168 N. J. A. Sloane (njas(AT)research.att.com).

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