A322176 Triangle read by rows: The number of chord diagrams on 2n vertices with m marked chords.

1, 1, 1, 2, 2, 2, 5, 8, 8, 5, 17, 39, 61, 39, 17, 79, 287, 556, 556, 287, 79, 554, 2792, 6910, 9058, 6910, 2792, 554, 5283, 34650, 103212, 171195, 171195, 103212, 34650, 5283, 65346, 510593, 1783325, 3559031, 4449494, 3559031, 1783325, 510593, 65346

Offset

0,4

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1274

R. J. Mathar, Marked Chord Diagrams A322176

R. J. Mathar, Feynman diagrams of the QED vacuum polarization, vixra:1901.0148 (2019)

Formula

T(n,m) = T(n,n-m).

T(n,0) = A054499(n).

Example

   1;
   1,   1;
   2,   2,   2;
   5,   8,   8,   5;
  17,  39,  61,  39,  17;
  79, 287, 556, 556, 287, 79;

Prog

(PARI)
C(p, d)={sum(k=0, p\2, binomial(p, 2*k) * (d*(1+y^d))^k * if(d%2, p==2*k, (1+y^(d/2))^(p-2*k)) * (2*k)!/(2^k*k!))}
R(n)={sum(k=0, n\2, binomial(n, 2*k) * (1+y^2)^k * (1+y)^(n-2*k) * (2*k)!/k!)}
row(n)={Vec(if(n==0, 1, (sumdiv(2*n, d, eulerphi(d)*C(2*n/d, d))/n + R(n) + (1+y)*R(n-1))/4))}
{ for(n=0, 8, print(row(n))) } \\ Andrew Howroyd, Dec 13 2018

Crossrefs

Cf. A054499.

Sequence in context: A162145 A374403 A208050 * A039886 A367087 A109523

Adjacent sequences: A322173 A322174 A322175 * A322177 A322178 A322179

Keyword

nonn,tabl

Author

R. J. Mathar, Nov 30 2018

Extensions

Terms a(21) and beyond from Andrew Howroyd, Dec 13 2018

Status

approved