A243474 Number of isoscent sequences of length n with exactly one descent.

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

Offset

3,2

Links

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 3..1000

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

Column k=1 of A242352.

Sequence in context: A061648 A281050 A267774 * A111644 A225618 A081278

Adjacent sequences: A243471 A243472 A243473 * A243475 A243476 A243477

Keyword

nonn

Author

Joerg Arndt and Alois P. Heinz, Jun 05 2014

Status

approved