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A213578 Antidiagonal sums of the convolution array A213576.

%I A213578 #32 May 09 2025 19:52:02
%S A213578 1,4,13,34,80,174,359,712,1371,2580,4768,8684,15629,27852,49225,86390,
%T A213578 150704,261530,451795,777360,1332791,2277864,3882048,6599064,11191705,
%U A213578 18940564,31992709,53943562,90807056,152631750,256190783
%N A213578 Antidiagonal sums of the convolution array A213576.
%H A213578 Clark Kimberling, <a href="/A213578/b213578.txt">Table of n, a(n) for n = 1..500</a>
%H A213578 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,-2,4,0,-1).
%F A213578 a(n) = 4*a(n-1) - 4*a(n-2) - 2*a(n-3) + 4*a(n-4) - a(n-6).
%F A213578 G.f.: (1 + x^2)/(1 - 2*x + x^3)^2.
%F A213578 a(n) = n*F(n+4) - 2*(F(n+5) - n - 5), F = A000045. - _Ehren Metcalfe_, Jul 05 2019
%t A213578 b[n_]:= n; c[n_]:= Fibonacci[n];
%t A213578 t[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
%t A213578 TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213578 Flatten[Table[t[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213576 *)
%t A213578 r[n_] := Table[t[n, k], {k,40}]  (* columns of antidiagonal triangle *)
%t A213578 d = Table[t[n, n], {n,1,40}] (* A213577 *)
%t A213578 s[n_] := Sum[t[i, n+1-i], {i, 1, n}]
%t A213578 s1 = Table[s[n], {n, 1, 50}] (* A213578 *)
%t A213578 (* alternate program *)
%t A213578 LinearRecurrence[{4,-4,-2,4,0,-1},{1,4,13,34,80,174},40] (* _Harvey P. Dale_, Jul 04 2019 *)
%o A213578 (Magma) [n*Fibonacci(n+4)-2*(Fibonacci(n+5)-n-5): n in [1..40]]; // _Vincenzo Librandi_, Jul 05 2019
%o A213578 (PARI) vector(40, n, n*fibonacci(n+4)-2*(fibonacci(n+5)-n-5)) \\ _G. C. Greubel_, Jul 05 2019
%o A213578 (Sage) [n*Fibonacci(n+4)-2*(Fibonacci(n+5)-n-5) for n in (1..40)] # _G. C. Greubel_, Jul 05 2019
%o A213578 (GAP) List([1..40], n-> n*Fibonacci(n+4)-2*(Fibonacci(n+5)-n-5)); # _G. C. Greubel_, Jul 05 2019
%Y A213578 Cf. A213576, A213500.
%K A213578 nonn,easy
%O A213578 1,2
%A A213578 _Clark Kimberling_, Jun 18 2012