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

1,2

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.

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

Example

x + 3*x^2 + 12*x^3 + 63*x^4 + 432*x^5 + 3798*x^6 + 41472*x^7 + 543483*x^8 + ...

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

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 *)

Prog

(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 */

Crossrefs

Cf. A005411.

Sequence in context: A135889 A369746 A124562 * A276743 A203508 A052757

Adjacent sequences: A172447 A172448 A172449 * A172451 A172452 A172453

Keyword

nonn

Author

N. J. A. Sloane, Nov 20 2010

Status

approved