A212344 - OEIS

A212344

Sequence of coefficients of x^(n-3) in marked mesh pattern generating function Q_{n,132}^(0,3,0,0)(x).

%I #40 Mar 26 2022 03:38:16

%S 5,5,10,25,70,210,660,2145,7150,24310,83980,293930,1040060,3714500,

%T 13372200,48474225,176788350,648223950,2388193500,8836315950,

%U 32820602100,122331335100,457412818200,1715298068250,6449520736620,24309732007260,91836765360760

%N Sequence of coefficients of x^(n-3) in marked mesh pattern generating function Q_{n,132}^(0,3,0,0)(x).

%C Sequence appears to be 5*A000108 (with a different offset). - _Peter Bala_, Nov 26 2013

%H S. Kitaev, J. Remmel and M. Tiefenbruck, <a href="http://arxiv.org/abs/1201.6243">Marked mesh patterns in 132-avoiding permutations I</a>, arXiv preprint arXiv:1201.6243, 2012

%F Conjecture: (n-2)*a(n) = 2*(2n-7)*a(n-1). _R. J. Mathar_, Jun 27 2012

%F G.f.: conjecture: 5*T(0), where T(k) = 1 - x/( x - 1/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Nov 26 2013

%F G.f.: conjecture: 5*(1-sqrt(1-4*x))/(2*x) = 10/T(0), where T(k) = 2*x*(2*k+1) + k+2 - 2*x*(k+2)*(2*k+3)/T(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Nov 26 2013

%p A212344List := proc(m) local A, P, n; A := [5,5]; P := [5];

%p for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-1]]);

%p A := [op(A), P[-1]] od; A end: A212344List(27); # _Peter Luschny_, Mar 26 2022

%t QQ0[t, x] = ( (1 - (1-4*x*t)^(1/2)) ) / (2*x*t); QQ1[t, x] = 1/(1 - t*QQ0[t, x]); QQ2[t, x] = (1 + t*(QQ1[t, x] - QQ0[t, x]))/(1 - t*QQ0[t, x]); QQ3[t, x] = (1 + t*(QQ2[t, x] - QQ0[t, x] + t*(QQ1[t, x] - QQ0[t, x])))/(1 - t*QQ0[t, x]); Simplify[Series[QQ3[t, x], {t, 0, 35}]] (* _Robert Price_, Jun 03 2012 *)

%K nonn

%O 3,1

%A _N. J. A. Sloane_, May 09 2012

%E a(11)-a(36) from _Robert Price_, Jun 03 2012