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A213580 Principal diagonal of the convolution array A213579.

%I A213580 #22 Nov 11 2024 12:59:57
%S A213580 1,5,15,35,74,146,277,511,925,1651,2916,5108,8889,15385,26507,45491,
%T A213580 77806,132678,225645,382835,648121,1095075,1846920,3109800,5228209,
%U A213580 8777261,14716167,24643331,41220050,68873786,114964741,191719783
%N A213580 Principal diagonal of the convolution array A213579.
%H A213580 Clark Kimberling, <a href="/A213580/b213580.txt">Table of n, a(n) for n = 1..1000</a>
%H A213580 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,-2,4,0,-1).
%F A213580 a(n) = 4*a(n-1) - 4*a(n-2) - 2*a(n-3) + 4*a(n-4) + a(n-5).
%F A213580 G.f.: x*(1 + x - x^2 - 3*x^3)/(1 - 2*x + x^3)^2.
%F A213580 a(n) = Fibonacci(n+3) + n*Fibonacci(n+2) - 2*(n+1). - _G. C. Greubel_, Jul 08 2019
%t A213580 (* First program *)
%t A213580 b[n_]:= Fibonacci[n]; c[n_]:= n;
%t A213580 T[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
%t A213580 TableForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213580 Flatten[Table[T[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213579 *)
%t A213580 r[n_]:= Table[T[n, k], {k, 40}]
%t A213580 d = Table[T[n, n], {n, 1, 40}] (* A213580 *)
%t A213580 s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
%t A213580 s1 = Table[s[n], {n, 1, 50}] (* A053808 *)
%t A213580 (* Second program *)
%t A213580 Table[Fibonacci[n+3] + n*Fibonacci[n+2] -2*(n+1), {n, 40}] (* _G. C. Greubel_, Jul 08 2019 *)
%o A213580 (PARI) vector(40, n, f=fibonacci; f(n+3) +n*f(n+2) -2*(n+1)) \\ _G. C. Greubel_, Jul 08 2019
%o A213580 (Magma) F:=Fibonacci; [F(n+3) + n*F(n+2) -2*(n+1): n in [1..40]]; // _G. C. Greubel_, Jul 08 2019
%o A213580 (Sage) f=fibonacci; [f(n+3) +n*f(n+2) -2*(n+1) for n in (1..40)] # _G. C. Greubel_, Jul 08 2019
%o A213580 (GAP) F:=Fibonacci;; List([1..40], n-> F(n+3) +n*F(n+2) -2*(n+1)); # _G. C. Greubel_, Jul 08 2019
%Y A213580 Cf. A000045, A213500, A213579.
%K A213580 nonn,easy
%O A213580 1,2
%A A213580 _Clark Kimberling_, Jun 18 2012