A243228 Number of isoscent sequences of length n with exactly two ascents.

3, 25, 128, 525, 1901, 6371, 20291, 62407, 187272, 552104, 1606762, 4631643, 13256644, 37742047, 107025452, 302585780, 853556449, 2403702976, 6760469822, 18995826302, 53336990264, 149680752886, 419883986837, 1177504825907, 3301408010791, 9254726751126

Offset

4,1

Links

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 4..1000

Formula

Recurrence: (3*n^3 - 43*n^2 + 120*n + 20)*a(n) = (21*n^3 - 289*n^2 + 712*n + 400)*a(n-1) - (51*n^3 - 665*n^2 + 1374*n + 1540)*a(n-2) + 4*(12*n^3 - 145*n^2 + 230*n + 435)*a(n-3) - (9*n^3 - 87*n^2 + 26*n + 280)*a(n-4) - 2*(3*n^3 - 34*n^2 + 43*n + 100)*a(n-5). - Vaclav Kotesovec, Aug 27 2014

a(n) ~ c * d^n, where d = 2.8019377358048382524722... is the root of the equation 1 + 3*d - 4*d^2 + d^3 = 0, c = 0.9786935821895919379992... is the root of the equation 1 - 49*c^2 + 49*c^3 = 0. - Vaclav Kotesovec, Aug 27 2014

G.f.: x^4*(3 - 5*x + x^2)*(1 - x - x^2) / ((1 - x)^3*(1 - 2*x)^2*(1 - 4*x + 3*x^2 + x^3)) (conjectured). - Colin Barker, May 05 2019

Maple

b:= proc(n, i, t) option remember; `if`(n<1, 1, expand(add(
      `if`(j>i, x, 1) *b(n-1, j, t+`if`(j=i, 1, 0)), j=0..t+1)))
    end:
a:= n-> coeff(b(n-1, 0$2), x, 2):
seq(a(n), n=4..35);

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n < 1, 1, Expand[Sum[ If[j > i, x, 1] *b[n - 1, j, t + If[j == i, 1, 0]], {j, 0, t + 1}]]]; a[n_] := Coefficient [b[n - 1, 0, 0], x, 2]; Table[a[n], {n, 4, 35}] (* Jean-François Alcover, Feb 09 2015, after Maple *)

Crossrefs

Column k=2 of A242351.

Sequence in context: A095664 A215773 A099868 * A112495 A034578 A265874

Adjacent sequences: A243225 A243226 A243227 * A243229 A243230 A243231

Keyword

nonn

Author

Joerg Arndt and Alois P. Heinz, Jun 01 2014

Status

approved