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A213584 Rectangular array: (row n) = b**c, where b(h) = F(h+1), c(h) = n-1+h, where F=A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.

%I A213584 #15 Sep 08 2022 08:46:02
%S A213584 1,4,2,10,7,3,21,16,10,4,40,32,22,13,5,72,59,43,28,16,6,125,104,78,54,
%T A213584 34,19,7,212,178,136,97,65,40,22,8,354,299,231,168,116,76,46,25,9,585,
%U A213584 496,386,284,200,135,87,52,28,10,960,816,638,473,337,232,154,98,58,31,11
%N A213584 Rectangular array:  (row n) = b**c, where b(h) = F(h+1), c(h) = n-1+h, where F=A000045 (Fibonacci numbers), n>=1, h>=1, and **  = convolution.
%C A213584 Principal diagonal: A213585.
%C A213584 Antidiagonal sums: A213586.
%C A213584 Row 1, (1,2,3,5,...)**(1,2,3,4,...): A001891.
%C A213584 Row 2, (1,2,3,5,...)**(2,3,4,5,...): A023550.
%C A213584 Row 3, (1,2,3,5,...)**(3,4,5,6,...): A023554.
%C A213584 For a guide to related arrays, see A213500.
%H A213584 Clark Kimberling, <a href="/A213584/b213584.txt">Antidiagonals n = 1..60</a>
%F A213584 T(n,k) = 3*T(n,k-1) - 2*T(n,k-2) - T(n,k-3) + T(n,k-4).
%F A213584 G.f. for row n:  f(x)/g(x), where f(x) = n + x - (n - 1)*x and g(x) = (1 - x - x^2)*(1 - x)^2.
%F A213584 T(n, k) = Fibonacci(k+4) + n*Fibonacci(k+3) - 2*(n+k) - 3. - _G. C. Greubel_, Jul 08 2019
%e A213584 Northwest corner (the array is read by falling antidiagonals):
%e A213584 1...4....10...21...40....72
%e A213584 2...7....16...32...59....104
%e A213584 3...10...22...43...78....136
%e A213584 4...13...28...54...97....168
%e A213584 5...16...34...65...116...200
%e A213584 6...19...40...76...135...232
%t A213584 (* First program *)
%t A213584 b[n_]:= Fibonacci[n+1]; c[n_]:= n;
%t A213584 T[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
%t A213584 TableForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213584 Flatten[Table[T[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213584 *)
%t A213584 r[n_]:= Table[T[n, k], {k, 40}]  (* columns of antidiagonal triangle *)
%t A213584 d = Table[T[n, n], {n, 1, 40}] (* A213585 *)
%t A213584 s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
%t A213584 s1 = Table[s[n], {n, 1, 50}] (* A213586 *)
%t A213584 (* Second program *)
%t A213584 Table[Fibonacci[n-k+5] + k*Fibonacci[n-k+4] -2*n-5, {n, 12}, {k, n}]//Flatten (* _G. C. Greubel_, Jul 08 2019 *)
%o A213584 (PARI) t(n,k) = fibonacci(n-k+5) + k*fibonacci(n-k+4) -(2*n+5);
%o A213584 for(n=1,12, for(k=1,n, print1(t(n,k), ", "))) \\ _G. C. Greubel_, Jul 08 2019
%o A213584 (Magma) [[Fibonacci(n-k+5) + k*Fibonacci(n-k+4) -(2*n+5): k in [1..n]]: n in [1..12]]; // _G. C. Greubel_, Jul 08 2019
%o A213584 (Sage) [[fibonacci(n-k+5) + k*fibonacci(n-k+4) -(2*n+5) for k in (1..n)] for n in (1..12)] # _G. C. Greubel_, Jul 08 2019
%o A213584 (GAP) Flat(List([1..12], n-> List([1..n], k-> Fibonacci(n-k+5) + k*Fibonacci(n-k+4) -(2*n+5)))) # _G. C. Greubel_, Jul 08 2019
%Y A213584 Cf. A000045, A213500, A213579.
%K A213584 nonn,tabl,easy
%O A213584 1,2
%A A213584 _Clark Kimberling_, Jun 18 2012