A215341 - OEIS

%I #34 Jul 29 2023 10:28:03

%S 1,0,0,0,1,1,1,1,5,10,16,23,53,118,232,411,813,1718,3568,7012,13925,

%T 28603,59533,121878,247915,509136,1057278,2194138,4536943,9394145,

%U 19552639,40803472,85131237,177640486,371426592,778275264,1632420197,3425607187,7195476245,15134138683,31866093569

%N Expansion of series_reversion( x/(1+x^4*sum(k>=0, x^k)) ) / x.

%C Number of Dyck words of semilength n with substrings UUU...UU (ascents) only of lengths >= 4. See A215340 for an explanation. [_Joerg Arndt_, Apr 16 2013]

%H Alois P. Heinz, <a href="/A215341/b215341.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f. A(x) satisfies 0 = -x^4*A(x)^4 - x*A(x)^2 + (x + 1)*A(x) - 1. [_Joerg Arndt_, Mar 01 2014]

%F Recurrence: 2*n*(n+1)*(2*n+3)*(16204*n^4 - 82948*n^3 + 139973*n^2 - 85643*n + 10674)*a(n) =  - (n-1)*n*(307876*n^5 - 960260*n^4 + 288863*n^3 + 582749*n^2 + 5406*n + 12696)*a(n-1) + 4*(n-2)*(129632*n^6 - 469136*n^5 + 354226*n^4 + 317255*n^3 - 469674*n^2 + 176517*n - 21420)*a(n-2) - 2*(n-3)*(n-2)*(16204*n^5 - 34336*n^4 + 82943*n^3 - 208775*n^2 + 192120*n - 40656)*a(n-3) + 6*(n-3)*(n-2)*(97224*n^5 - 351852*n^4 + 179198*n^3 + 540009*n^2 - 571727*n + 92968)*a(n-4) + 229*(n-4)*(n-3)*(n-2)*(16204*n^4 - 18132*n^3 - 11647*n^2 + 10275*n - 1740)*a(n-5). - _Vaclav Kotesovec_, Mar 22 2014

%F a(n) ~ sqrt((s-1)*s^3/(6-8*s+3*s^2)) / (2*sqrt(Pi)*n^(3/2)*r^n), where r = 0.4577644245749322..., s = 1.232809919151165... are roots of the system of equations 1 + r*s^2 + r^4*s^4 = (1+r)*s, 1+r = 2*r*s + 4*r^4*s^3. - _Vaclav Kotesovec_, Mar 22 2014

%F a(n) = (1/(n+1)) * Sum_{i=0..floor(n/4)} C(n+1,i) * C(n-3*i-1,n-4*i). - _Vladimir Kruchinin_, Apr 01 2019

%p b:= proc(x, y, t) option remember; `if`(y<x, 0, `if`(y=0,

%p       `if`(t in [0, 4], 1, 0), `if`(x>0 and t in [0, 4],

%p        b(x-1, y, 0), 0) +b(x, y-1, min(t+1, 4))))

%p     end:

%p a:= n-> b(n, n, 0):

%p seq(a(n), n=0..50);  # _Alois P. Heinz_, Apr 16 2013

%t InverseSeries[x/(1+x^4/(1-x)) + O[x]^50] // CoefficientList[#, x]& // Rest (* _Jean-François Alcover_, Mar 29 2017 *)

%o (PARI) N=66; Vec( serreverse(x/(1+x^4*sum(k=0,N,x^k))+O(x^N)) / x )

%o (Maxima)

%o a(n):=sum(binomial(n+1,i)*binomial(n-3*i-1,n-4*i),i,0,floor(n/4))/(n+1); /* _Vladimir Kruchinin_, Apr 01 2019 */

%Y Cf. A000108 (rev. of x/(1+1*sum(k>=1,x^k)) ), A005043 (rev. of x/(1+x*sum(k>=1,x^k)) ), A114997 (rev. of x/(1+x^2*sum(k>=1,x^k)) ).

%Y Cf. A001003 (rev. of x*(1-1*sum(k=1,N,x^k)) ), A046736 (rev. of x*(1-x*sum(k=1,N,x^k)) ), A054514 (rev. of x*(1-x^2*sum(k=1,N,x^k)) ), A215342 (rev. of x*(1-x^3*sum(k=1,N,x^k)) ).

%K nonn

%O 0,9

%A _Joerg Arndt_, Aug 19 2012

%E Modified definition to obtain offset 0 for combinatorial interpretation, _Joerg Arndt_, Apr 16 2013