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A213585 Principal diagonal of the convolution array A213584.

%I A213585 #17 Sep 08 2022 08:46:02
%S A213585 1,7,22,54,116,232,443,821,1490,2664,4710,8256,14373,24883,42878,
%T A213585 73594,125880,214664,365087,619425,1048666,1771852,2988362,5031744,
%U A213585 8459401,14201887,23811238,39873726,66695420,111440104,186016835
%N A213585 Principal diagonal of the convolution array A213584.
%H A213585 Clark Kimberling, <a href="/A213585/b213585.txt">Table of n, a(n) for n = 1..1000</a>
%H A213585 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,-2,4,0,-1).
%F A213585 a(n) = 4*a(n-1) - 4*a(n-2) - 2*a(n-3) + 4*a(n-4) + a(n-5).
%F A213585 G.f.: x*(1 + 3*x - 2*x^2 - 4*x^3 - 2*x^4)/(1 - 2*x + x^3)^2.
%F A213585 a(n) = Fibonacci(n+4) + n*Fibonacci(n+3) - (4*n + 3). - _G. C. Greubel_, Jul 08 2019
%t A213585 (* First program *)
%t A213585 b[n_]:= Fibonacci[n+1]; c[n_]:= n;
%t A213585 T[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
%t A213585 TableForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213585 Flatten[Table[T[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213584 *)
%t A213585 r[n_]:= Table[T[n, k], {k, 40}]  (* columns of antidiagonal triangle *)
%t A213585 d = Table[T[n, n], {n, 1, 40}] (* A213585 *)
%t A213585 s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
%t A213585 s1 = Table[s[n], {n, 1, 50}] (* A213586 *)
%t A213585 (* Second program *)
%t A213585 Table[Fibonacci[n+4] + n*Fibonacci[n+3] -4*n-3, {n, 40}] (* _G. C. Greubel_, Jul 08 2019 *)
%o A213585 (PARI) vector(40, n, f=fibonacci; f(n+4) +n*f(n+3) -(4*n+3)) \\ _G. C. Greubel_, Jul 08 2019
%o A213585 (Magma) F:=Fibonacci; [F(n+4) +n*F(n+3) -(4*n+3): n in [1..40]]; // _G. C. Greubel_, Jul 08 2019
%o A213585 (Sage) f=fibonacci; [f(n+4) +n*f(n+3) -(4*n+3) for n in (1..40)] # _G. C. Greubel_, Jul 08 2019
%o A213585 (GAP) F:=Fibonacci;; List([1..40], n-> F(n+4) +n*F(n+3) -(4*n+3)) # _G. C. Greubel_, Jul 08 2019
%Y A213585 Cf. A000045, A213500, A213584.
%K A213585 nonn,easy
%O A213585 1,2
%A A213585 _Clark Kimberling_, Jun 18 2012