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A005412
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Feynman diagrams of order 2n.
(Formerly M3050)
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2
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1, 3, 18, 153, 1638, 20898, 307908, 5134293, 95518278, 1961333838, 44069970348, 1075902476058, 28367410077468, 803551902237828, 24342558819042888, 785445178323709773, 26896354975287884358, 974297972094661642518
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| There was a typo in the value of a(10) = 1967333838 previously given in the database (taken from the self-energies column of Table 1 in P. Cvitanovic et al.). The corrected value is given above. - Peter Bala, Mar 07 2011.
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REFERENCES
| P. Cvitanovic, B. Lautrup and R. B. Pearson, Number and weights of Feynman diagrams, Phys. Rev. D 18 (1978), 1939-1949.
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).
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LINKS
| P. Cvitanovic, B. Lautrup and R. B. Pearson, The number and weights of Feynman diagrams, Phys. Rev. D18, 1939 (1978).
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FORMULA
| See recurrence in Martin-Kearney paper.
From Peter Bala, Mar 07 2011: (Start)
The o.g.f. A(x) = x^2+3*x^4+18*x^6+153*x^8+ ... satisfies the differential equation A(x) = x^2+x^3*A'(x)+A(x)^2 (equation 3.55, P. Cvitanovic et al., A'(x) the derivative of A(x)).
Conjectural o.g.f. as a continued fraction:
x^2/(1-3*x^2/(1-3*x^2/(1-5*x^2/(1-5*x^2/(1-7*x^2/(1-7*x^2/(1-...))))))).
(End).
a(n) = (2*n - 2) * a(n-1) + Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 23 2011
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EXAMPLE
| x + 3*x^2 + 18*x^3 + 153*x^4 + 1638*x^5 + 20898*x^6 + 307908*x^7 + ...
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PROG
| (PARI) {a(n) = local(A); if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (2*k - 2) * A[k-1] + sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 23 2011 */
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CROSSREFS
| Sequence in context: A138420 A152409 A200320 * A145350 A107888 A138274
Adjacent sequences: A005409 A005410 A005411 * A005413 A005414 A005415
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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