A212344 Sequence of coefficients of x^(n-3) in marked mesh pattern generating function Q_{n,132}^(0,3,0,0)(x).

5, 5, 10, 25, 70, 210, 660, 2145, 7150, 24310, 83980, 293930, 1040060, 3714500, 13372200, 48474225, 176788350, 648223950, 2388193500, 8836315950, 32820602100, 122331335100, 457412818200, 1715298068250, 6449520736620, 24309732007260, 91836765360760

Offset

3,1

Comments

Sequence appears to be 5*A000108 (with a different offset). - Peter Bala, Nov 26 2013

Links

Table of n, a(n) for n=3..29.

S. Kitaev, J. Remmel and M. Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv preprint arXiv:1201.6243, 2012

Formula

Conjecture: (n-2)*a(n) = 2*(2n-7)*a(n-1). R. J. Mathar, Jun 27 2012

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

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

Maple

A212344List := proc(m) local A, P, n; A := [5, 5]; P := [5];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-1]]);
A := [op(A), P[-1]] od; A end: A212344List(27); # Peter Luschny, Mar 26 2022

Mathematica

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 *)

Crossrefs

Sequence in context: A245418 A082450 A304266 * A298181 A290332 A302842

Adjacent sequences: A212341 A212342 A212343 * A212345 A212346 A212347

Keyword

nonn

Author

N. J. A. Sloane, May 09 2012

Extensions

a(11)-a(36) from Robert Price, Jun 03 2012

Status

approved