A213589 - OEIS

%I #20 Sep 08 2022 08:46:02

%S 1,6,20,55,135,308,668,1395,2830,5610,10914,20904,39515,73860,136720,

%T 250937,457137,827260,1488190,2662905,4741946,8407236,14846100,

%U 26120400,45801925,80064018,139553708,242597035,420678315,727792580

%N Antidiagonal sums of the convolution array A213587.

%H Clark Kimberling, <a href="/A213589/b213589.txt">Table of n, a(n) for n = 1..500</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-5,0,3,1).

%F a(n) = 3*a(n-1) - 5*a(n-3) + 3*a(n-5) + a(n-6).

%F G.f.: x*(1 + 3*x + 2*x^2)/(1 - x - x^2)^3.

%F a(n) = (n+1)/2*Sum_{j=0..(n+1)/2}((n-j)*binomial(n-j+1,j)). - _Vladimir Kruchinin_, Apr 09 2016

%F a(n) = (n+1)*(n*Lucas(n+3) - 2*Fibonacci(n))/10 = (n+1)*((n+2) *Fibonacci(n+3) + 2*(n-2)*Fibonacci(n+2))/10. - _G. C. Greubel_, Jul 08 2019

%t (* First program *)

%t b[n_]:= Fibonacci[n+1]; c[n_]:= Fibonacci[n+1];

%t T[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]

%t TableForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]]

%t Flatten[Table[T[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213587 *)

%t r[n_]:= Table[T[n, k], {k, 40}]  (* columns of antidiagonal triangle *)

%t Table[T[n, n], {n, 1, 40}] (* A213588 *)

%t s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]

%t Table[s[n], {n, 1, 50}] (* A213589 *)

%t (* Second program *)

%t Table[(n+1)*(n*LucasL[n+3] -2*Fibonacci[n])/10, {n, 35}]] (* _G. C. Greubel_, Jul 08 2019 *)

%o (Maxima)

%o a(n):=(n+1)/2*sum((n-j)*binomial(n-j+1,j),j,0,(n+1)/2); /* _Vladimir Kruchinin_, Apr 09 2016 */

%o (PARI) vector(35, n, f=fibonacci; (n+1)*((n+2)*f(n+3)+ 2*(n-2)*f(n+2) )/10) \\ _G. C. Greubel_, Jul 08 2019

%o (Magma) F:=Fibonacci; [(n+1)*((n+2)*F(n+3) + 2*(n-2)*F(n+2))/10: n in [1..35]]; // _G. C. Greubel_, Jul 08 2019

%o (Sage) f=fibonacci; [(n+1)*((n+2)*f(n+3)+ 2*(n-2)*f(n+2) )/10 for n in (1..35)] # _G. C. Greubel_, Jul 08 2019

%o (GAP) F:=Fibonacci;; List([1..35], n-> (n+1)*((n+2)*F(n+3) + 2*(n-2)*F(n+2))/10) # _G. C. Greubel_, Jul 08 2019

%Y Cf. A213587, A213500.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Jun 19 2012