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A005411 Feynman diagrams of order 2n with exact propagators.
(Formerly M3610)
3
1, 4, 25, 208, 2146, 26368, 375733, 6092032, 110769550, 2232792064, 49426061818, 1192151302144, 31123028996164, 874428204384256, 26308967412122125, 843984969276915712, 28757604639850111894, 1037239628039528906752, 39481325230750749160462 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Cvitanovic et al. paper relates this sequence to A000698 and A005413 - Robert Munafo, Jan 24 2010

(x + 4x^2 + 25x^3 + 208x^4 + ...) = (x + 2x^2 + 7x^3 + 38x^4 + ...) * 1/(1 + x + 2x^2 + 7x^3 + 38x^4 + ...); where A094664 = (1, 1, 2, 7, 38, 286,...). - Gary W. Adamson, Nov 16 2011.

The Martin and Kearney article has S(2,-4,1) = [1,1,4,25,...] where u_1 = u_2 = 1, u_3 = 4, u_4 =25, etc. This is almost the same as this sequence.

REFERENCES

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

LINKS

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. D 18, 1939-1949 (1978).

R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, arXiv:1103.4936 [math.CO]

R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318. see p. 294.

FORMULA

From Peter Bala, Mar 07 2011: (Start)

Given o.g.f. A(x), then function F(x) := 1 + A(x^2) 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. A(x) as a continued fraction:

x/(1-4*x-3^2*x^2/(1-8*x-5^2*x^2/(1-12*x-7^2*x^2/(1-16*x-...)))).

Asymptotics: a(n) ~ 1/Pi*2^(n+1)*n!*(1-1/(2*n)-3/(8*n^2)).

(End)

Given u(1) = 1, u(n) = (2*n - 4) * u(n-1) + Sum_{k=1..n-1} u(k) * u(n-k) when n>1, then a(n) = u(n+1) if n>0. - 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

G.f.: 1/x^2 - 1/x - G(0)/(2*x^2), where G(k)= 1 + 1/(1 - 2*x*(2*k+1)/(2*x*(2*k+1) - 1 + 2*x*(2*k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 29 2013

G.f.: W(0)/x - 1/x, where W(k) = 1 - x*(2*k+1)/( x*(2*k+1) - 1/(1 - x*(2*k+3)/( x*(2*k+3) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 26 2013

G.f.: G(0)/x -1/x, where G(k) = 1 - x*(2*k+1)/(x - 1/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 21 2014

EXAMPLE

G.f. = x + 4*x^2 + 25*x^3 + 208*x^4 + 2146*x^5 + 26368*x^6 + 375733*x^7 + ...

MATHEMATICA

a[n_] := Module[{A}, A[1] = 1; A[k_] := A[k] = (2*k-4)*A[k-1]+Sum[A[j]*A[k-j], {j, 1, k-1}]; A[n]]; Table[a[n], {n, 2, 20}] (* Jean-Fran├žois Alcover, Feb 27 2014, after Michael Somos *)

a[ n_] := Module[{m = n + 1, u}, If[ n < 2, Boole[n == 1], u = Range[m]; Do[ u[[k]] = (2 k - 4) u[[k - 1]] + Sum[ u[[j]] u[[k - j]], {j, k - 1}], {k, 2, m}]; u[[m]]]]; (* Michael Somos, Feb 27 2014 *)

PROG

(PARI) {a(n) = local(A); if( n<1, 0, n++; 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 */

CROSSREFS

Cf. A094664.

Sequence in context: A036242 A120955 A061714 * A105628 A203219 A064299

Adjacent sequences:  A005408 A005409 A005410 * A005412 A005413 A005414

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Name corrected by Charles R Greathouse IV, Jan 24 2014

STATUS

approved

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Last modified July 23 23:07 EDT 2014. Contains 244873 sequences.