A027056 - OEIS

%I #15 Mar 08 2023 04:15:12

%S 0,2,4,8,18,42,102,256,658,1722,4570,12264,33212,90626,248892,687360,

%T 1907506,5316266,14873082,41751944,117567784,331979650,939807344,

%U 2666718976,7583071868,21605822594,61672362872,176338826728,505001067346,1448365610778,4159725843526,11962301199744

%N a(n) = A027052(n, 2n-1).

%H G. C. Greubel, <a href="/A027056/b027056.txt">Table of n, a(n) for n = 1..750</a>

%F Conjecture:b D-finite with recurrence (-n+1)*a(n) +2*(3*n-4)*a(n-1) +(-7*n+3)*a(n-2) +4*(-2*n+13)*a(n-3) +(5*n-29)*a(n-4) +2*(n-2)*a(n-5) +3*(n-5)*a(n-6)=0. - _R. J. Mathar_, Jun 15 2020

%F a(n) ~ 3^(n + 3/2) / (2 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Mar 08 2023

%p T:= proc(n, k) option remember;

%p       if k<0 or k>2*n then 0

%p     elif k=0 or k=2 or k=2*n then 1

%p     elif k=1 then 0

%p     else add(T(n-1, k-j), j=1..3)

%p       fi

%p     end:

%p seq( T(n,2*n-1), n=1..30); # _G. C. Greubel_, Nov 06 2019

%t T[n_, k_]:= T[n, k]= If[k<0 || k>2*n, 0, If[k==0 || k==2 || k==2*n, 1, If[k==1, 0, Sum[T[n-1, k-j], {j, 3}]]]]; Table[T[n,2*n-1], {n,30}] (* _G. C. Greubel_, Nov 06 2019 *)

%o (Sage)

%o @CachedFunction

%o def T(n, k):

%o     if (k<0 or k>2*n): return 0

%o     elif (k==0 or k==2 or k==2*n): return 1

%o     elif (k==1): return 0

%o     else: return sum(T(n-1, k-j) for j in (1..3))

%o [T(n,2*n-1) for n in (1..30)] # _G. C. Greubel_, Nov 06 2019

%K nonn

%O 1,2

%A _Clark Kimberling_

%E More terms from _Sean A. Irvine_, Oct 22 2019