A213589 Antidiagonal sums of the convolution array A213587.

1, 6, 20, 55, 135, 308, 668, 1395, 2830, 5610, 10914, 20904, 39515, 73860, 136720, 250937, 457137, 827260, 1488190, 2662905, 4741946, 8407236, 14846100, 26120400, 45801925, 80064018, 139553708, 242597035, 420678315, 727792580

Offset: 1

Author

Clark Kimberling, Jun 19 2012

Keywords

• nonn
• easy

Links

• Clark Kimberling, Table of n, a(n) for n = 1..500
• Index entries for linear recurrences with constant coefficients, signature (3,0,-5,0,3,1).

Crossrefs

Cf. A213587, A213500.

Programs

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

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

Mathematica

(* First program *)
b[n_]:= Fibonacci[n+1]; c[n_]:= Fibonacci[n+1];
T[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
TableForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[T[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213587 *)
r[n_]:= Table[T[n, k], {k, 40}]  (* columns of antidiagonal triangle *)
Table[T[n, n], {n, 1, 40}] (* A213588 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213589 *)
(* Second program *)
Table[(n+1)*(n*LucasL[n+3] -2*Fibonacci[n])/10, {n, 35}] (* G. C. Greubel, Jul 08 2019 *)

Maxima

a(n):=(n+1)/2*sum((n-j)*binomial(n-j+1,j),j,0,(n+1)/2); /* Vladimir Kruchinin, Apr 09 2016 */

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

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

Formula

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

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

a(n) = (n+1)/2*Sum_{j=0..(n+1)/2}((n-j)*binomial(n-j+1,j)). - Vladimir Kruchinin, Apr 09 2016

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