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A213504 Principal diagonal of the convolution array A213590.

%I A213504 #23 May 09 2025 23:08:35
%S A213504 1,6,35,138,488,1564,4733,13734,38711,106846,290496,781264,2084753,
%T A213504 5531846,14619811,38527834,101328712,266119228,698218525,1830665830,
%U A213504 4797572551,12568780126,32920653120,86214096768,225758326273
%N A213504 Principal diagonal of the convolution array A213590.
%H A213504 Clark Kimberling, <a href="/A213504/b213504.txt">Table of n, a(n) for n = 1..1000</a>
%H A213504 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (6,-10,-2,15,-2,-8,0,1).
%F A213504 a(n) = 6*a(n-1) - 10*a(n-2) - 2*a(n-3) + 15*a(n-4) - 2*a(n-5)- 8*a(n-6) + a(n-8).
%F A213504 G.f.: x*(1 + 9*x^2 - 10*x^3 + 7*x^4 - 2*x^5)/((1 - 3*x + x^2)*(1 - x - x^2)^3). [corrected by _Georg Fischer_, May 11 2019]
%F A213504 a(n) = Fibonacci(2*n+6) - Fibonacci(n+6) - 2*n*Fibonacci(n+3) - n^2*Fibonacci(n+1). - _G. C. Greubel_, Jul 06 2019
%t A213504 (* First program *)
%t A213504 b[n_]:= n^2; c[n_]:= Fibonacci[n];
%t A213504 T[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
%t A213504 TableForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213504 Flatten[Table[T[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213590 *)
%t A213504 r[n_]:= Table[T[n, k], {k, 40}]  (* columns of antidiagonal triangle *)
%t A213504 Table[T[n, n], {n, 1, 40}] (* A213504 *)
%t A213504 s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
%t A213504 Table[s[n], {n, 1, 50}] (* A213557 *)
%t A213504 (* Second program *)
%t A213504 With[{F = Fibonacci}, Table[F[2*n+6] -F[n+6] -2*n*F[n+3] -n^2*F[n+1], {n, 40}]] (* _G. C. Greubel_, Jul 06 2019 *)
%o A213504 (PARI) vector(40, n, my(f=fibonacci); f(2*n+6) - f(n+6) - 2*n*f(n+3) - n^2*f(n+1)) \\ _G. C. Greubel_, Jul 06 2019
%o A213504 (Magma) F:=Fibonacci; [F(2*n+6) -F(n+6) -2*n*F(n+3) -n^2*F(n+1): n in [1..40]]; // _G. C. Greubel_, Jul 06 2019
%o A213504 (Sage) f=fibonacci; [f(2*n+6) -f(n+6) -2*n*f(n+3) -n^2*f(n+1) for n in (1..40)] # _G. C. Greubel_, Jul 06 2019
%o A213504 (GAP) F:=Fibonacci;; List([1..40], n-> F(2*n+6) -F(n+6) -2*n*F(n+3) -n^2*F(n+1)); # _G. C. Greubel_, Jul 06 2019
%Y A213504 Cf. A213590, A213500.
%K A213504 nonn,easy
%O A213504 1,2
%A A213504 _Clark Kimberling_, Jun 19 2012