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A213557 Antidiagonal sums of the convolution array A213590.

%I A213557 #18 Jun 04 2025 19:08:31
%S A213557 1,6,23,70,184,438,971,2042,4125,8076,15424,28876,53189,96670,173747,
%T A213557 309362,546456,958690,1672015,2901170,5011321,8621976,14781888,
%U A213557 25263000,43053769,73186038,124119311,210055582,354806200,598245006
%N A213557 Antidiagonal sums of the convolution array A213590.
%H A213557 Clark Kimberling, <a href="/A213557/b213557.txt">Table of n, a(n) for n = 1..500</a>
%H A213557 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (5,-8,2,6,-4,-1,1).
%F A213557 a(n) = 5*a(n-1) - 8*a(n-2) + 2*a(n-3) + 6*a(n-4) - 4*a(n-5) - a(n-6) + a(n-7).
%F A213557 G.f.: f(x)/g(x), where f(x) = x*(1 + x + x^2 + x^3) and g(x) = (1 - x)^3 (1 - x - x^2)^2.
%F A213557 a(n) = n*Fibonacci(n+7) - 2*Fibonacci(n+9) + 2*n^2 + 20*n + 68. - _G. C. Greubel_, Jul 06 2019
%t A213557 (* First program *)
%t A213557 b[n_]:= n^2; c[n_]:= Fibonacci[n];
%t A213557 T[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
%t A213557 TableForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213557 Flatten[Table[T[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213590 *)
%t A213557 r[n_]:= Table[T[n, k], {k, 40}]  (* columns of antidiagonal triangle *)
%t A213557 Table[T[n, n], {n, 1, 40}] (* A213504 *)
%t A213557 s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
%t A213557 Table[s[n], {n, 1, 50}] (* A213557 *)
%t A213557 (* Second program *)
%t A213557 With[{F = Fibonacci}, Table[n*F[n+7] -2*F[n+9] +2*(n^2+10*n+34), {n,40}]] (* _G. C. Greubel_, Jul 06 2019 *)
%t A213557 LinearRecurrence[{5,-8,2,6,-4,-1,1},{1,6,23,70,184,438,971},30] (* _Harvey P. Dale_, Jun 04 2025 *)
%o A213557 (PARI) vector(40, n, f=fibonacci; n*f(n+7) -2*f(n+9) +2*(n^2+10*n+34)) \\ _G. C. Greubel_, Jul 06 2019
%o A213557 (Magma) F:=Fibonacci; [n*F(n+7) -2*F(n+9) +2*(n^2+10*n+34): n in [1..40]]; // _G. C. Greubel_, Jul 06 2019
%o A213557 (Sage) f=fibonacci; [n*f(n+7) -2*f(n+9) +2*(n^2+10*n+34) for n in (1..40)] # _G. C. Greubel_, Jul 06 2019
%o A213557 (GAP) F:=Fibonacci;; List([1..40], n-> n*F(n+7) -2*F(n+9) +2*(n^2+10*n+ 34)); # _G. C. Greubel_, Jul 06 2019
%Y A213557 Cf. A213590, A213500.
%K A213557 nonn,easy
%O A213557 1,2
%A A213557 _Clark Kimberling_, Jun 19 2012