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A005411
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Feynman diagrams of order 2n.
(Formerly M3610)
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3
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1, 4, 25, 208, 2146, 26368, 375733, 6092032, 110769550, 2232792064, 49426061818, 1192151302144, 31123028996164, 874428204384256, 26308967412122125, 843984969276915712, 28757604639850111894, 1037239628039528906752, 39481325230750749160462
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OFFSET
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1,2
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COMMENTS
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Cvitanovic et al. paper relates this sequence to A000698 and A005413 [From Robert Munafo, Jan 24 2010]
(1 + 4x + 25x^2 + 208x^3 + ...) = (1 + 2x + 7x^2 + 38x^3 + ...) * 1/(1 + x + 2x^2 + 7x^3 + 38x^4 + ...); where A094664 = (1, 2, 7, 38, 286,...). - Gary W. Adamson, Nov 16 2011.
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REFERENCES
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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. 294.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Table of n, a(n) for n=1..19.
P. Cvitanovic, B. Lautrup and R. B. Pearson, The number and weights of Feynman diagrams, Phys. Rev. D18, 1939 (1978).
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence
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FORMULA
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See recurrence in Martin-Kearney paper.
From Peter Bala, Mar 07 2011: (Start)
O.g.f.: A(x) = sum {n = 1..inf} a(n)*x^(2*n) = x^2 + 4*x^4 + 25*x^6 + 208*x^8 + ....
The function F(x) := 1 + A(x) satisfies the differential equation F(x) = 1 + x^3*d/dx(F(x)) + x^2*F(x)^2 (equation 3.53, P. Cvitanovic et al.).
Conjectural o.g.f. as a continued fraction:
x^2/(1-4*x^2-3^2*x^4/(1-8*x^2-5^2*x^4/(1-12*x^2-7^2*x^4/(1-16*x^2-...)))).
Asymptotics: a(n) ~ 1/Pi*2^(n+1)*n!*(1-1/(2*n)-3/(8*n^2)).
(End)
a(n) = (2*n - 4) * a(n-1) + Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 24 2011
G.f.: 1/Q(0) where Q(k) = 1 - x*(2*k+1)/(1 - x*(2*k+3)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 19 2013
G.f.: 1/x^2 - 1/x - Q(0)/x^2, where Q(k)= 1 - x*(2*k+1)/(1 - x*(2*k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 20 2013
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EXAMPLE
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x + x^2 + 4*x^3 + 25*x^4 + 208*x^5 + 2146*x^6 + 26368*x^7 + 375733*x^8 + ...
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PROG
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(PARI) {a(n) = local(A); if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n,
A[k] = (2 * k - 4) * A[k-1] + sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 24 2011 */
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CROSSREFS
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Cf. A094664
Sequence in context: A036242 A120955 A061714 * A105628 A203219 A064299
Adjacent sequences: A005408 A005409 A005410 * A005412 A005413 A005414
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KEYWORD
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nonn,easy,changed
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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