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

%I A243228 #23 May 05 2019 07:56:14
%S A243228 3,25,128,525,1901,6371,20291,62407,187272,552104,1606762,4631643,
%T A243228 13256644,37742047,107025452,302585780,853556449,2403702976,
%U A243228 6760469822,18995826302,53336990264,149680752886,419883986837,1177504825907,3301408010791,9254726751126
%N A243228 Number of isoscent sequences of length n with exactly two ascents.
%H A243228 Joerg Arndt and Alois P. Heinz, <a href="/A243228/b243228.txt">Table of n, a(n) for n = 4..1000</a>
%F A243228 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
%F A243228 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
%F A243228 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
%p A243228 b:= proc(n, i, t) option remember; `if`(n<1, 1, expand(add(
%p A243228       `if`(j>i, x, 1) *b(n-1, j, t+`if`(j=i, 1, 0)), j=0..t+1)))
%p A243228     end:
%p A243228 a:= n-> coeff(b(n-1, 0$2), x, 2):
%p A243228 seq(a(n), n=4..35);
%t A243228 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 *)
%Y A243228 Column k=2 of A242351.
%K A243228 nonn
%O A243228 4,1
%A A243228 _Joerg Arndt_ and _Alois P. Heinz_, Jun 01 2014