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

Author

Joerg Arndt and Alois P. Heinz, Jun 01 2014

Keywords

• nonn

Links

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

Crossrefs

Column k=2 of A242351.

Programs

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 *)

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