Search: id:A003169
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%I A003169 M2973
%S A003169 1,3,14,79,494,3294,22952,165127,1217270,9146746,69799476,539464358,
%T A003169 4214095612,33218794236,263908187100,2110912146295,16985386737830,
%U A003169 137394914285538,1116622717709012,9113225693455362,74659999210200292
%N A003169 Number of 2-line arrays; or number of P-graphs with 2n edges.
%C A003169 First column of triangle A100326. - Paul D. Hanna (pauldhanna(AT)juno.com), 
               Nov 16 2004
%D A003169 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A003169 M. Bicknell and V. E. Hoggatt, Jr., Sequences of matrix inverses from 
               Pascal, Catalan and related convolution arrays, Fib. Quart., 14 (1976), 
               224-232.
%D A003169 L. Carlitz, Enumeration of two-line arrays, Fib. Quart., 11 (1973), 113-130.
%D A003169 R. C. Read, On the enumeration of a class of plane multigraphs, Aequat. 
               Math., 31 (1986), 47-63.
%H A003169 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=416">
               Encyclopedia of Combinatorial Structures 416</a>
%F A003169 For formula see Read reference.
%F A003169 a(n) = ( (324*n^2-708*n+360)*a(n-1) - (371*n^2-1831*n+2250)*a(n-2) + 
               (20*n^2-130*n+210)*a(n-3) )/(16*n*(2*n-1)) for n>2, with a(0)=0, 
               a(1)=1, a(2)=3. - Paul D. Hanna (pauldhanna(AT)juno.com), Nov 16 
               2004
%F A003169 G.f. satisfies: A(x) = x*(1+A(x))/(1-A(x))^2 where A(0)=0. G.f. satisfies: 
               (1+A(x))/(1-A(x)) = 2*G003168(x)-1, where G003168 is the g.f. of 
               A003168. - Paul D. Hanna (pauldhanna(AT)juno.com), Nov 16 2004
%F A003169 a(n) = (1/n)*Sum_{i=0..n-1} binomial(n,i)*binomial(3*n-i-2,n-i-1). - 
               Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 13 2006
%p A003169 a[0]:=0:a[1]:=1:a[2]:=3:for n from 3 to 30 do a[n]:=((324*n^2-708*n+360)*a[n-1] 
               -(371*n^2-1831*n+2250)*a[n-2]+(20*n^2-130*n+210)*a[n-3])/(16*n*(2*n-1)) 
               od:seq(a[n],n=1..25); (Deutsch)
%o A003169 (PARI) {a(n)=if(n==0,0,if(n==1,1,if(n==2,3,( (324*n^2-708*n+360)*a(n-1) 
               -(371*n^2-1831*n+2250)*a(n-2)+(20*n^2-130*n+210)*a(n-3))/(16*n*(2*n-1)) 
               )))} (Hanna)
%o A003169 (PARI) {a(n)=local(A=x+x*O(x^n));if(n==1,1, for(i=1,n,A=x*(1+A)/(1-A)^2); 
               polcoeff(A,n))}
%Y A003169 Cf. A003168, A100324, A100326.
%Y A003169 Sequence in context: A074538 A001564 A059276 this_sequence A086621 A020089 
               A027614
%Y A003169 Adjacent sequences: A003166 A003167 A003168 this_sequence A003170 A003171 
               A003172
%K A003169 nonn,easy
%O A003169 1,2
%A A003169 N. J. A. Sloane (njas(AT)research.att.com).
%E A003169 More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 31 2005

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