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%I M4259 #38 Jan 31 2022 06:48:00
%S 1,1,6,50,518,6354,89782,1435330,25625910,505785122,10944711398,
%T 257834384850,6572585595622,180334118225650,5300553714899094,
%U 166206234856979810,5538980473666776854,195527829569946627138,7288988096561232432070
%N Vertex diagrams of order 2n.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A005416/b005416.txt">Table of n, a(n) for n = 0..100</a>
%H D. J. Broadhurst, <a href="https://arxiv.org/abs/hep-ph/9909336">Four-loop Dyson-Schwinger-Johnson anatomy</a>, arXiv:hep-ph/9909336, 1999.
%H P. Cvitanovic, <a href="https://doi.org/10.1016/0550-3213(77)90357-1">Asymptotic estimates and gauge invariance</a>, Nuclear Phys. B 127 (1977), 176-188.
%H R. J. Martin and M. J. Kearney, <a href="http://arXiv.org/abs/1103.4936">An exactly solvable self-convolutive recurrence</a>, arXiv:1103.4936 [math.CO], 2011.
%H R. J. Martin and M. J. Kearney, <a href="http://dx.doi.org/10.1007/s00010-010-0051-0">An exactly solvable self-convolutive recurrence</a>, Aequat. Math., 80 (2010), 291-318. see p. 292.
%F 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).
%F a(n) = (2*n - 1) * A000698(n). [Martin and Kearney]
%e G.f. = 1 + x + 6*x^2 + 50*x^3 + 518*x^4 + 6354*x^5 + 89782*x^6 + 1435330*x^7 + ...
%t 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. *)
%o (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 */
%o (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 */
%Y Cf. A000698, A001147, A049464.
%K nonn,nice,easy
%O 0,3
%A _N. J. A. Sloane_
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