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A243228
Number of isoscent sequences of length n with exactly two ascents.
2
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: A215773 A377555 A099868 * A112495 A034578 A265874
KEYWORD
nonn
AUTHOR
Joerg Arndt and Alois P. Heinz, Jun 01 2014
STATUS
approved