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Search: id:A100861
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| A100861 |
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Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n). |
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12
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| 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, 36, 378, 1260, 945, 1, 45, 630, 3150, 4725, 945, 1, 55, 990, 6930, 17325, 10395, 1, 66, 1485, 13860, 51975, 62370, 10395, 1, 78, 2145, 25740, 135135, 270270, 135135, 1, 91, 3003, 45045, 315315, 945945, 945945, 135135
(list; graph; listen)
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OFFSET
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0,6
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COMMENT
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Row n contains 1+floor(n/2) terms. Row sums yield A000085. T(2n,n)=T(2n-1,n-1)=(2n-1)!! (A001147).
Inverse binomial transform is triangle with T(2n,n)=(2n-1)!!, 0 otherwise. - Paul Barry (pbarry(AT)wit.ie), May 21 2005
Equivalently, number of involutions of n with k pairs. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jun 09 2006
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REFERENCES
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J. Y. Choi and J. D. H. Smith, On the unimodilty and combinatorics of Bessel numbers, Discrete Math., 264 (2003), 45-53.
C. D. Godsil, Algebraic Combinatorics, Chapman & Hall, New York, 1993.
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FORMULA
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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].
T(n, k)=binomial(n, 2k)(2k-1)!! - Paul Barry (pbarry(AT)wit.ie), May 21 2002. [Corrected by Roland Hildebrand, Mar 06 2009]
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
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EXAMPLE
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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}.
Triangle starts:
1;
1;
1,1;
1,3;
1,6,3;
1,10,15;
1,15,45,15;
...
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MAPLE
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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
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PROGRAM
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(PARI) T(n, k)=if(k<0|2*k>n, 0, n!/k!/(n-2*k)!/2^k) /* Michael Somos Jun 04 2005 */
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CROSSREFS
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Other versions of this same triangle are given in A144299, A001497, A001498, A111924.
Cf. A000085 (row sums).
Sequence in context: A131110 A133093 A065567 this_sequence A131031 A130452 A133085
Adjacent sequences: A100858 A100859 A100860 this_sequence A100862 A100863 A100864
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KEYWORD
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nonn,tabf,nice
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 08 2005
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