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

Author

Joerg Arndt and Alois P. Heinz, Jun 05 2014

Keywords

• nonn

Examples

			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].
		

Links

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

Crossrefs

Column k=1 of A242352.

Programs

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n<1, 1, Expand[Sum[If[jJean-François Alcover, Feb 09 2015, after A242352 *)

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