cp's OEIS Frontend

A213774 Rectangular array: (row n) = b**c, where b(h) = F(h+1), c(h) = 2*n-3+2*h, F=A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.

%I A213774 #21 Jul 10 2019 09:06:25
%S A213774 1,5,3,14,11,5,31,26,17,7,61,53,38,23,9,112,99,75,50,29,11,197,176,
%T A213774 137,97,62,35,13,337,303,240,175,119,74,41,15,566,511,409,304,213,141,
%U A213774 86,47,17,939,850,685,515,368,251,163,98,53,19,1545,1401,1134
%N A213774 Rectangular array:  (row n) = b**c, where b(h) = F(h+1), c(h) = 2*n-3+2*h, F=A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.
%C A213774 Principal diagonal:  A213775.
%C A213774 Antidiagonal sums:  A213776.
%C A213774 Row 1,  (1,2,3,5,8,...)**(1,3,5,7,9,...): A023652.
%C A213774 Row 2,  (1,2,3,5,8,...)**(3,5,7,9,11,...).
%C A213774 Row 3,  (1,2,3,5,8,...)**(5,7,9,11,13,...).
%C A213774 For a guide to related arrays, see A213500.
%H A213774 Clark Kimberling, <a href="/A213774/b213774.txt">Antidiagonals n=1..60, flattened</a>
%F A213774 T(n,k) = 3*T(n,k-1)-2*T(n,k-2)-T(n,k-3)+T(n,k-4).
%F A213774 G.f. for row n:  f(x)/g(x), where f(x) = x*(2*n - 1 + 2*x - (2*n - 3)*x^2) and g(x) = (1 - x - x^2)*(1 - x )^2.
%F A213774 T(n,k) = 2*n*Fibonacci(k+3) + Lucas(k+3) - 4*(k+n+1). - _Ehren Metcalfe_, Jul 08 2019
%e A213774 Northwest corner (the array is read by falling antidiagonals):
%e A213774 1....5....14...31....61....112
%e A213774 3....11...26...53....99....176
%e A213774 5....17...38...75....137...240
%e A213774 7....23...50...97....175...304
%e A213774 9....29...62...119...213...368
%e A213774 11...35...74...141...251...432
%t A213774 b[n_] := Fibonacci[n + 1]; c[n_] := 2 n - 1;
%t A213774 t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
%t A213774 TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
%t A213774 Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
%t A213774 r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213774 *)
%t A213774 Table[t[n, n], {n, 1, 40}] (* A213775 *)
%t A213774 s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
%t A213774 Table[s[n], {n, 1, 50}] (* A213776 *)
%Y A213774 Cf. A023652, A213500, A213768, A213775, A213776.
%K A213774 nonn,tabl,easy
%O A213774 1,2
%A A213774 _Clark Kimberling_, Jun 21 2012