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A213567 Principal diagonal of the convolution array A213566.

%I A213567 #24 Sep 08 2022 08:46:02
%S A213567 1,13,59,183,476,1108,2409,4993,10007,19559,37504,70832,132145,244029,
%T A213567 446763,811847,1465676,2630836,4697945,8350305,14779671,26058903,
%U A213567 45784224,80179968,139995361,243755533,423324539,733409943
%N A213567 Principal diagonal of the convolution array A213566.
%H A213567 Clark Kimberling, <a href="/A213567/b213567.txt">Table of n, a(n) for n = 1..1000</a>
%H A213567 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (6,-12,5,12,-12,-3,6,0,-1).
%F A213567 a(n) = 6*a(n-1) - 12*a(n-2) - 5*a(n-3) + 12*a(n-4) - 12*a(n-5) - 3*a(n-6) + 6*a(n-7) - a(n-9).
%F A213567 G.f.:  f(x)/g(x), where f(x) = x*(1 + 7*x - 7*x^2 - 20*x^3 + 9*x^4 + 9*x^5 + 9*x^6) and g(x) = (1 - 2*x + x^3)^3.
%F A213567 a(n) = (2*n + 3)*Fibonacci(n+3) + (n^2 + 2)*Fibonacci(n+2) - 4*(n^2 + 2*n + 2). - _G. C. Greubel_, Jul 26 2019
%t A213567 (* First program *)
%t A213567 b[n_]:= Fibonacci[n]; c[n_]:= n^2;
%t A213567 t[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
%t A213567 TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213567 Flatten[Table[t[n-k+1, k], {n, 12}, {k, n, 1, -1}]]
%t A213567 r[n_]:= Table[t[n, k], {k, 1, 60}]  (* A213566 *)
%t A213567 d = Table[t[n, n], {n, 1, 40}] (* A213567 *)
%t A213567 s[n_]:= Sum[t[i, n+1-i], {i, 1, n}]
%t A213567 s1 = Table[s[n], {n, 1, 50}] (* A213570 *)
%t A213567 (* Second program *)
%t A213567 Table[(2*n+3)*Fibonacci[n+3] +(n^2+2)*Fibonacci[n+2] -4*(n^2+2*n+2), {n, 30}] (* _G. C. Greubel_, Jul 26 2019 *)
%o A213567 (PARI) vector(30, n, f=fibonacci; (2*n+3)*f(n+3)+(n^2+2)*f(n+2) -4*(n^2+ 2*n+2)) \\ _G. C. Greubel_, Jul 26 2019
%o A213567 (Magma) F:= Fibonacci; [(2*n+3)*F(n+3)+(n^2+2)*F(n+2) -4*(n^2+2*n+2): n in [1..30]]; // _G. C. Greubel_, Jul 26 2019
%o A213567 (Sage) f=fibonacci; [(2*n+3)*f(n+3)+(n^2+2)*f(n+2) -4*(n^2+ 2*n+2) for n in (1..30)] # _G. C. Greubel_, Jul 26 2019
%o A213567 (GAP) F:=Fibonacci;; List([1..30], n-> (2*n+3)*F(n+3)+(n^2+2)*F(n+2) -4*(n^2+2*n+2)); # _G. C. Greubel_, Jul 26 2019
%Y A213567 Cf. A213566, A213500.
%K A213567 nonn,easy
%O A213567 1,2
%A A213567 _Clark Kimberling_, Jun 19 2012