A212343 a(n) = (n+1)*(n-2)*(n-3)/2.

0, 0, 5, 18, 42, 80, 135, 210, 308, 432, 585, 770, 990, 1248, 1547, 1890, 2280, 2720, 3213, 3762, 4370, 5040, 5775, 6578, 7452, 8400, 9425, 10530, 11718, 12992, 14355, 15810, 17360, 19008, 20757, 22610, 24570, 26640, 28823, 31122, 33540, 36080, 38745, 41538, 44462, 47520, 50715, 54050, 57528

Offset: 2

Author

N. J. A. Sloane, May 09 2012

Keywords

• nonn
• easy

Comments

Sequence of coefficients of x^1 in marked mesh pattern generating function Q_{n,132}^(0,3,0,0)(x).

Is this row 2 of the convolution array A213819? - Clark Kimberling, Jul 04 2012

Links

• Colin Barker, Table of n, a(n) for n = 2..1000
• S. Kitaev, J. Remmel and M. Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv preprint arXiv:1201.6243 [math.CO], 2012.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Partial sums are in A241765.

Cf. similar sequences of the type m*(m+1)*(m+k)/2 listed in A267370.

Cf. also A212342.

Programs

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]); CoefficientList[Coefficient[Simplify[Series[QQ3[t, x], {t, 0, 35}]],x],t]  (* Robert Price, Jun 04 2012 *)
LinearRecurrence[{4,-6,4,-1},{0,0,5,18},60] (* Harvey P. Dale, Mar 15 2018 *)

PARI

Vec(-x^4*(2*x-5)/(x-1)^4 + O(x^100)) \\ Colin Barker, Jul 10 2015

Formula

For n>=4, a(n) = (n-3)*A212342(n-1).

a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>7. - Colin Barker, Jul 10 2015

G.f.: -x^4*(2*x-5) / (x-1)^4. - Colin Barker, Jul 10 2015

From Amiram Eldar, Apr 03 2022: (Start)

Sum_{n>=4} 1/a(n) = 23/72.

Sum_{n>=4} (-1)^n/a(n) = 4*log(2)/3 - 55/72. (End)

Extensions

a(10)-a(35) from Robert Price, Jun 02 2012

Entry revised by N. J. A. Sloane, Sep 10 2016