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A322176 Triangle read by rows: The number of chord diagrams on 2n vertices with m marked chords. 2
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 (list; table; graph; refs; listen; history; text; internal format)
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: A039878 A162145 A208050 * A039886 A109523 A008295

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

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Last modified July 7 14:14 EDT 2020. Contains 335495 sequences. (Running on oeis4.)