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%I #14 Mar 16 2024 10:49:38
%S 1,6,29,124,499,1933,7307,27166,99841,363980,1319404,4763927,17155264,
%T 61672791,221499015,795198010,2854898575,10253237150,36846414395,
%U 132518215788,477049025009,1719101735260,6201858101192,22399768386170,80998670324341,293244129636085
%N Number of isoscent sequences of length n with exactly one descent.
%H Joerg Arndt and Alois P. Heinz, <a href="/A243474/b243474.txt">Table of n, a(n) for n = 3..1000</a>
%F Recurrence: 2*(n+1)*(n+2)*(2*n+3)*(12397*n^7 - 189057*n^6 + 1186699*n^5 - 4027875*n^4 + 7966576*n^3 - 8920548*n^2 + 4726368*n - 164160)*a(n) = 2*(n+1)*(2*n + 1)*(74382*n^8 - 997975*n^7 + 5169531*n^6 - 12939205*n^5 + 13804539*n^4 + 4032932*n^3 - 23655372*n^2 + 20014848*n - 1477440)*a(n-1) - 2*(347116*n^9 - 3797013*n^8 + 13426236*n^7 - 10697862*n^6 - 45689304*n^5 + 115458855*n^4 - 47561392*n^3 - 85364460*n^2 + 57311424*n - 1477440)*a(n-2) - (1822359*n^10 - 28485611*n^9 + 180879786*n^8 - 605859318*n^7 + 1141835871*n^6 - 1127112699*n^5 + 285267312*n^4 + 592120604*n^3 - 783881808*n^2 + 409889664*n - 50388480)*a(n-3) - 2*(247940*n^10 - 5628293*n^9 + 49022694*n^8 - 212356554*n^7 + 479773884*n^6 - 459074385*n^5 - 250049558*n^4 + 1020252416*n^3 - 830684880*n^2 + 423719136*n - 432293760)*a(n-4) + 2*(n-4)*(2070299*n^9 - 30481583*n^8 + 181205557*n^7 - 572698754*n^6 + 1060137133*n^5 - 1157719883*n^4 + 582047111*n^3 + 378941580*n^2 - 897279300*n + 403878960)*a(n-5) + 6*(n-5)*(n-4)*(768614*n^8 - 9316516*n^7 + 43281239*n^6 - 101853145*n^5 + 131895047*n^4 - 75435871*n^3 - 41445228*n^2 + 118112292*n - 101468592)*a(n-6) + 117*(n-6)*(n-5)*(n-4)*(12397*n^7 - 102278*n^6 + 312694*n^5 - 496340*n^4 + 374821*n^3 + 103402*n^2 - 440568*n + 590400)*a(n-7). - _Vaclav Kotesovec_, Jun 06 2014
%F a(n) ~ c * d^n / n^(3/2), where d = 1/6*(847+33*sqrt(33))^(1/3) + 44/(3*(847+33*sqrt(33))^(1/3)) + 2/3 = 3.802619145513318... is the root of the equation 4*d^3 - 8*d^2 - 24*d - 13 = 0 and c = sqrt(2565 + 2*(3*(692007507 - 5151139*sqrt(33)))^(1/3) + 2*(3*(692007507 + 5151139*sqrt(33)))^(1/3)) / (4*sqrt(21*Pi)) = 2.695007157151120689163873119078514352395445402... . - _Vaclav Kotesovec_, Jun 06 2014, updated Mar 16 2024
%e a(4) = 6: [0,0,1,0], [0,0,2,0], [0,0,2,1], [0,1,0,0], [0,1,0,1], [0,1,1,0].
%t 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, 1]; Table[a[n], {n, 3, 40}] (* _Jean-François Alcover_, Feb 09 2015, after A242352 *)
%Y Column k=1 of A242352.
%K nonn
%O 3,2
%A _Joerg Arndt_ and _Alois P. Heinz_, Jun 05 2014
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