Search: id:A100861
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%I A100861
%S A100861 1,1,1,1,1,3,1,6,3,1,10,15,1,15,45,15,1,21,105,105,1,28,210,420,105,1,
%T A100861 36,378,1260,945,1,45,630,3150,4725,945,1,55,990,6930,17325,10395,1,66,
%U A100861 1485,13860,51975,62370,10395,1,78,2145,25740,135135,270270,135135,1,91,
               3003,45045,315315,945945,945945,135135
%N A100861 Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings 
               of the complete graph K(n).
%C A100861 Row n contains 1+floor(n/2) terms. Row sums yield A000085. T(2n,n)=T(2n-1,
               n-1)=(2n-1)!! (A001147).
%C A100861 Inverse binomial transform is triangle with T(2n,n)=(2n-1)!!, 0 otherwise. 
               - Paul Barry (pbarry(AT)wit.ie), May 21 2005
%C A100861 Equivalently, number of involutions of n with k pairs. - Frank Adams-Watters 
               (FrankTAW(AT)Netscape.net), Jun 09 2006
%D A100861 J. Y. Choi and J. D. H. Smith, On the unimodilty and combinatorics of 
               Bessel numbers, Discrete Math., 264 (2003), 45-53.
%D A100861 C. D. Godsil, Algebraic Combinatorics, Chapman & Hall, New York, 1993.
%F A100861 T(n, k)=n!/[k!(n-2k)!*2^k]. E.g.f.=exp(z+tz^2/2). G.f.=g(t, z) satisfies 
               the differential equation g=1+zg+tz^2*diff(zg, z). Row generating 
               polynomial=P[n]=[ -i*sqrt(t/2)]^n*H(n, i/sqrt(2t)), where H(n, x) 
               is a Hermite polynomial and i=sqrt(-1). Row generating polynomials 
               P[n] satisfy P[0]=1, P[n]=P[n-1]+(n-1)tP[n-2].
%F A100861 T(n, k)=binomial(n, 2k)(2k-1)!! - Paul Barry (pbarry(AT)wit.ie), May 
               21 2002. [Corrected by Roland Hildebrand, Mar 06 2009]
%F A100861 T(n,k) = (n-2k+1)T(n-1,k-1) + T(n-1,k). - Frank Adams-Watters (FrankTAW(AT)Netscape.net), 
               Jun 09 2006
%e A100861 T(4,2)=3 because in the graph with vertex set {A,B,C,D} and edge set 
               {AB,BC,CD,AD,AC,BD} we have the following three 2-matchings: {AB,
               CD},{AC,BD} and {AD,BC}.
%e A100861 Triangle starts:
%e A100861 1;
%e A100861 1;
%e A100861 1,1;
%e A100861 1,3;
%e A100861 1,6,3;
%e A100861 1,10,15;
%e A100861 1,15,45,15;
%e A100861 ...
%p A100861 P[0]:=1: for n from 1 to 14 do P[n]:=sort(expand(P[n-1]+(n-1)*t*P[n-2])) 
               od: for n from 0 to 14 do seq(coeff(t*P[n],t^k),k=1..1+floor(n/2)) 
               od; # yields the sequence in triangular form
%o A100861 (PARI) T(n,k)=if(k<0|2*k>n, 0, n!/k!/(n-2*k)!/2^k) /* Michael Somos Jun 
               04 2005 */
%Y A100861 Other versions of this same triangle are given in A144299, A001497, A001498, 
               A111924.
%Y A100861 Cf. A000085 (row sums).
%Y A100861 Sequence in context: A131110 A133093 A065567 this_sequence A131031 A130452 
               A133085
%Y A100861 Adjacent sequences: A100858 A100859 A100860 this_sequence A100862 A100863 
               A100864
%K A100861 nonn,tabf,nice
%O A100861 0,6
%A A100861 Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 08 2005

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