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 A005416 Vertex diagrams of order 2n. (Formerly M4259) 6
 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-François 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: A243667 A303562 A125558 * A300989 A105617 A297926 Adjacent sequences:  A005413 A005414 A005415 * A005417 A005418 A005419 KEYWORD nonn,nice,easy AUTHOR STATUS approved

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Last modified August 26 03:38 EDT 2019. Contains 326324 sequences. (Running on oeis4.)