A172450 The case S(2,0,-1) of the family of self-convolutive recurrences studied by Martin and Kearney.
1, 3, 12, 63, 432, 3798, 41472, 543483, 8301312, 144502218, 2818685952, 60826110678, 1437615931392, 36914181252588, 1022923413061632, 30419533530730323, 966125479213596672, 32634383834158752258, 1168128785923721920512
Offset: 1
Keywords
Examples
x + 3*x^2 + 12*x^3 + 63*x^4 + 432*x^5 + 3798*x^6 + 41472*x^7 + 543483*x^8 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 1..400
- A. V. Belitsky, G. P. Korchemsky, Octagon at finite coupling, arXiv:2003.01121 [hep-th], 2020.
- R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318. see p. 294.
- R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, arXiv:1103.4936 [math.CO], 2011.
Crossrefs
Cf. A005411.
Programs
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Maple
ser := series(BesselK(0,-1/x)/BesselK(1,-1/x) - 1, x, 20): seq((1/2)*4^n*coeff(ser,x,n), n=0..19); # Peter Luschny, Dec 11 2017
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Mathematica
Clear[a]; a[1] = 1; a[n_]:= a[n] = 2*n*a[n-1] - Sum[a[k]*a[n-k], {k, 1, n-1}]; Table[a[n], {n, 1, 20}] (* Vaclav Kotesovec, Jan 19 2015 *) -
PARI
{a(n) = local(A); if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = 2 * k * A[k-1] - sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 24 2011 */
Formula
a(n) = 2 * n * a(n-1) - Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 24 2011
G.f.: x / (1 - 3*x / (1 - 1*x / (1 - 5*x / (1 - 3*x / (1 - 7*x / (1 - 5*x / (1 - 9*x / (1 - 7*x / (1 - 11*x / (1 - 9*x / (1 - 13*x / (1 - 11*x / ... )))))))))))). - Michael Somos, Jan 03 2013
G.f.: (1/(2*Q(0) - 1) - 1)/(2*x) where Q(k) = 1 - x*(2*k+1)/( 1 - x*(2*k+1)/Q(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Apr 02 2013
G.f.: Q(0)/x - 1/x, where Q(k)= 1 - x*(2*k-1)/(1 - x*(2*k+3)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 21 2013
a(n) ~ 2^n * (n-1)! / Pi. - Vaclav Kotesovec, Jan 19 2015
a(n) = (1/2)*4^n*[x^n](BesselK(0,-1/x)/BesselK(1,-1/x) - 1). - Peter Luschny, Dec 11 2017