login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A005416 Vertex diagrams of order 2n.
(Formerly M4259)
4
1, 1, 6, 50, 518, 6354, 89782, 1435330, 25625910, 505785122, 10944711398, 257834384850, 6572585595622, 180334118225650, 5300553714899094, 166206234856979810, 5538980473666776854, 195527829569946627138, 7288988096561232432070 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

REFERENCES

P. Cvitanovic, Asymptotic estimates and gauge invariance, Nuclear Phys. B 127 (1977), 176-188.

R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318. see p. 293.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n=0..100

D. J. Broadhurst, Four-loop Dyson-Schwinger-Johnson anatomy

FORMULA

Let s_n=(2*n)!/(2^n*n!) (A001147), S(x)=Sum_{n >= 0} s_n*x^n; sequence has g.f. A(x) satisfying 1-1/S(x)=x*A(x)*S(x).

a(n) = (2*n - 1) * A000698(n). [Martin and Kearney]

EXAMPLE

G.f. = 1 + x + 6*x^2 + 50*x^3 + 518*x^4 + 6354*x^5 + 89782*x^6 + 1435330*x^7 + ...

MATHEMATICA

m = 19; s[x_] = Sum[(2*n)!/(2^n*n!)*x^n, {n, 0, m}]; gf[x_] = (s[x] - 1)/(s[x]^2*x); Most[CoefficientList[Series[gf[x], {x, 0, m}], x]] (* Jean-Fran├žois Alcover, Aug 31 2011, after g.f. *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = sum( k=0, n+1, (2*k)! / k! /2^k * x^k, x^2 * O(x^n)); polcoeff( (A - 1) / (x * A^2), n))}; /* Michael Somos, Oct 11 2006 */

(PARI) {a(n) = my(A); if( n<1, n==0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (2 * k - 3) * A[k-1] + sum( j=1, k-1, A[j] * A[k-j])); (2*n - 1) * A[n])}; /* Michael Somos, Jul 24 2011 */

CROSSREFS

Cf. A000698, A049464.

Sequence in context: A039742 A243667 A125558 * A105617 A094072 A058784

Adjacent sequences:  A005413 A005414 A005415 * A005417 A005418 A005419

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 4 15:24 EST 2016. Contains 278750 sequences.