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A213566 Rectangular array: (row n) = b**c, where b(h) = F(h), c(h) = (n-1+h)^2, F = A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.

%I A213566 #22 Sep 08 2022 08:46:02
%S A213566 1,5,4,15,13,9,36,33,25,16,76,71,59,41,25,148,140,120,93,61,36,273,
%T A213566 260,228,183,135,85,49,485,464,412,340,260,185,113,64,839,805,721,604,
%U A213566 476,351,243,145,81,1424,1369,1233,1044,836,636,456,309,181,100
%N A213566 Rectangular array:  (row n) = b**c, where b(h) = F(h), c(h) = (n-1+h)^2, F = A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.
%C A213566 Principal diagonal:  A213567.
%C A213566 Antidiagonal sums:  A213570.
%C A213566 Row 1,  (1,1,2,3,5,...)**(1,4,9,16,25,...): A053808.
%C A213566 Row 2,  (1,1,2,3,5,...)**(4,9,16,25,...).
%C A213566 Row 3,  (1,1,2,3,5,...)**(16,25,49,...).
%C A213566 For a guide to related arrays, see A213500.
%H A213566 Clark Kimberling, <a href="/A213566/b213566.txt">Antidiagonals n = 1..60, flattened</a>
%F A213566 T(n,k) = 4*T(n,k-1)-5*T(n,k-2)+T(n,k-3)+2*T(n,k-4)-T(n,k-5).
%F A213566 G.f. for row n:  f(x)/g(x), where f(x) = x*(n^2 - (2*n^2 - 2*n - 1)*x + (n - 1)^2 *x^2) and g(x) = (1 - x - x^2)*(1 - x )^3.
%F A213566 T(n,k) = n*(n*F(k+2) + 2*F(k+3)) + F(k+6) - (n+2)*(2*k+n+2) - k^2 - 4, F = A000045. - _Ehren Metcalfe_, Jul 10 2019
%e A213566 Northwest corner (the array is read by falling antidiagonals):
%e A213566 1....5....15....36....76
%e A213566 4....13...33....71....140
%e A213566 9....25...59....120...228
%e A213566 16...41...93....183...340
%e A213566 25...61...135...260...476
%t A213566 (* First program *)
%t A213566 b[n_]:= Fibonacci[n]; c[n_]:= n^2;
%t A213566 t[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
%t A213566 TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213566 Flatten[Table[t[n-k+1, k], {n, 12}, {k, n, 1, -1}]]
%t A213566 r[n_]:= Table[t[n, k], {k, 1, 60}]  (* A213566 *)
%t A213566 d = Table[t[n, n], {n, 1, 40}] (* A213567 *)
%t A213566 s[n_]:= Sum[t[i, n+1-i], {i, 1, n}]
%t A213566 s1 = Table[s[n], {n, 1, 50}] (* A213570 *)
%t A213566 (* Second program *)
%t A213566 With[{F = Fibonacci}, Table[k*(k*F[n-k+3] +2*F[n-k+4]) + F[n-k+7] -(k+2) *(2*n-k+4) -(n-k+1)^2 -4, {n, 12}, {k, n}]//Flatten] (* _G. C. Greubel_, Jul 26 2019 *)
%o A213566 (PARI) f=fibonacci;
%o A213566 for(n=1,12, for(k=1,n, print1(k*(k*f(n-k+3) +2*f(n-k+4)) + f(n-k+7) -(k+2)*(2*n-k+4) -(n-k+1)^2 -4, ", "))) \\ _G. C. Greubel_, Jul 26 2019
%o A213566 (Magma) F:=Fibonacci; [k*(k*F(n-k+3) +2*F(n-k+4)) + F(n-k+7) -(k+2)*(2*n-k+4) -(n-k+1)^2 -4: k in [1..n], n in [1..12]]; // _G. C. Greubel_, Jul 26 2019
%o A213566 (Sage) f=fibonacci; [[k*(k*f(n-k+3) +2*f(n-k+4)) + f(n-k+7) -(k+2)*(2*n-k+4) -(n-k+1)^2 -4 for k in (1..n)] for n in (1..12)] # _G. C. Greubel_, Jul 26 2019
%o A213566 (GAP) F:=Fibonacci;; Flat(List([1..12], n-> List([1..n], k-> k*(k*F(n-k+3) +2*F(n-k+4)) + F(n-k+7) -(k+2)*(2*n-k+4) -(n-k+1)^2 -4 ))); # _G. C. Greubel_, Jul 26 2019
%Y A213566 Cf. A213500, A213587.
%K A213566 nonn,tabl,easy
%O A213566 1,2
%A A213566 _Clark Kimberling_, Jun 19 2012