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

Author

R. J. Mathar, Nov 30 2018

Keywords

• nonn
• tabl

Examples

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

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)

Crossrefs

Cf. A054499.

Programs

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

Formula

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

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

Extensions

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