|
%I #25 Nov 07 2019 08:28:33
%S 1,4,13,38,111,326,961,2842,8425,25020,74403,221488,659895,1967422,
%T 5869055,17516540,52300729,156214828,466736979,1394894672,4169810935,
%U 12467680862,37285474803,111524444760,333633526937,998233861836
%N a(n) = Sum_{k=0..2n} (k+1) * A027052(n, 2n-k).
%C The terms a(0)..a(25) obey a linear recurrence with polynomial coefficients of degree 7. - _Ralf Stephan_, May 31 2014
%H G. C. Greubel, <a href="/A027076/b027076.txt">Table of n, a(n) for n = 0..1000</a>
%F 0 = a(n)*(9*n + 9) + a(n+1)*(3*n + 21) + a(n+2)*(13*n - 5) + a(n+3)*(-29*n + 11) + a(n+4)*(-13*n - 121) + a(n+5)*(25*n + 123) + a(n+6)*(-98n - 43) + a(n+7)*(n + 5) for n>=-1. - _Michael Somos_, May 31 2014
%F 0 = a(n)*(+81*a(n+1) + 189*a(n+2) + ... + 45*a(n+8)) + a(n+1)*(-135*a(n+1) + ...) + ... + a(n+7)*(-7*a(n+7) + a(n+8)) for n>=-1. - _Michael Somos_, May 31 2014
%e G.f. = 1 + 4*x + 13*x^2 + 38*x^3 + 111*x^4 + 326*x^5 + 961*x^6 + 2842*x^7 + ...
%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( add((k+1)*T(n,2*n-k), k=0..2*n), n=0..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[Sum[(k+1)*T[n,2*n-k], {k, 0, 2*n}], {n, 0, 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 [sum((k+1)*T(n,2*n-k) for k in (0..2*n)) for n in (0..30)] # _G. C. Greubel_, Nov 06 2019
%Y Cf. A027052.
%K nonn
%O 0,2
%A _Clark Kimberling_
|
