%I #18 Sep 08 2022 08:46:02
%S 1,5,1,15,6,2,36,20,11,3,76,51,35,17,5,148,112,87,55,28,8,273,224,188,
%T 138,90,45,13,485,421,372,300,225,145,73,21,839,758,694,596,488,363,
%U 235,118,34,1424,1324,1243,1115,968,788,588,380,191,55,2384,2263,2163,2001,1809,1564,1276,951,615,309,89
%N Rectangular array: (row n) = b**c, where b(h) = h^2, c(h) = F(n-1+h), F = A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.
%C Principal diagonal: A213504.
%C Antidiagonal sums: A213557.
%C Row 1, (1,4,9,16,...)**(1,1,2,3,5,...): A053808.
%C Row 2, (1,4,9,16,...)**(1,2,3,5,8,...): A213586.
%C Row 3, (1,4,9,16,...)**(2,3,5,8,13,...).
%C For a guide to related arrays, see A213500.
%H Clark Kimberling, <a href="/A213590/b213590.txt">Antidiagonals n = 1..60, flattened</a>
%F Rows: 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 Columns: T(n,k) = T(n-1,k) + T(n-2,k).
%F G.f. for row n: f(x)/g(x), where f(x) = F(n) + F(n+1)*x + F(n-1)*x^2 and g(x) = (1 - x - x^2)*(1 - x )^3.
%F T(n, k) = Fibonacci(n+k+6) - Fibonacci(n+6) - 2*k*Fibonacci(n+3) - k^2*Fibonacci(n+1). - _G. C. Greubel_, Jul 05 2019
%e Northwest corner (the array is read by falling antidiagonals):
%e 1....5....15....36....76.....148
%e 1....6....20....51....112....224
%e 2....11...35....87....188....372
%e 3....17...55....138...300....596
%e 5....28...90....225...488....868
%e 8....45...145...363...788....1564
%e 13...73...235...588...1276...2532
%t (* First program *)
%t b[n_]:= n^2; c[n_]:= Fibonacci[n];
%t T[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
%t TableForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]]
%t Flatten[Table[T[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213590 *)
%t r[n_]:= Table[T[n, k], {k, 40}] (* columns of antidiagonal triangle *)
%t Table[T[n, n], {n, 1, 40}] (* A213504 *)
%t s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
%t Table[s[n], {n, 1, 50}] (* A213557 *)
%t (* Second program *)
%t t[n_, k_]:= Fibonacci[n+7] - Fibonacci[k+6] - 2*(n-k+1)*Fibonacci[k+3] - (n-k+1)^2*Fibonacci[k+1]; Table[t[n, k], {n, 1, 12}, {k, 1, n}]//Flatten (* _G. C. Greubel_, Jul 05 2019 *)
%o (PARI) f=fibonacci; t(n,k) = f(n+7) -f(k+6) -2*(n-k+1)*f(k+3) -(n-k+1)^2 *f(k+1);
%o for(n=1,12, for(k=1,n, print1(t(n,k), ", "))) \\ _G. C. Greubel_, Jul 05 2019
%o (Magma) F:=Fibonacci; [[F(n+7) -F(k+6) -2*(n-k+1)*F(k+3) -(n-k+1)^2 *F(k+1): k in [1..n]]: n in [1..12]]; // _G. C. Greubel_, Jul 05 2019
%o (Sage) f=fibonacci; [[f(n+7) -f(k+6) -2*(n-k+1)*f(k+3) - (n-k+1)^2* f(k+1) for k in (1..n)] for n in (1..12)] # _G. C. Greubel_, Jul 05 2019
%o (GAP) F:=Fibonacci;; Flat(List([1..12],n-> List([1..n],k-> F(n+7)-F(k+6) -2*(n-k+1)*F(k+3)-(n-k+1)^2*F(k+1) ))) # _G. C. Greubel_, Jul 05 2019
%Y Cf. A213500, A213587.
%K nonn,tabl,easy
%O 1,2
%A _Clark Kimberling_, Jun 19 2012