|
| |
|
|
A243474
|
|
Number of isoscent sequences of length n with exactly one descent.
|
|
2
|
|
|
|
1, 6, 29, 124, 499, 1933, 7307, 27166, 99841, 363980, 1319404, 4763927, 17155264, 61672791, 221499015, 795198010, 2854898575, 10253237150, 36846414395, 132518215788, 477049025009, 1719101735260, 6201858101192, 22399768386170, 80998670324341, 293244129636085
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
3,2
|
|
|
LINKS
|
|
|
|
FORMULA
|
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
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
|
|
|
EXAMPLE
|
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].
|
|
|
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, 1]; Table[a[n], {n, 3, 40}] (* Jean-François Alcover, Feb 09 2015, after A242352 *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
