A213557 Antidiagonal sums of the convolution array A213590.

1, 6, 23, 70, 184, 438, 971, 2042, 4125, 8076, 15424, 28876, 53189, 96670, 173747, 309362, 546456, 958690, 1672015, 2901170, 5011321, 8621976, 14781888, 25263000, 43053769, 73186038, 124119311, 210055582, 354806200, 598245006

Offset: 1

Author

Clark Kimberling, Jun 19 2012

Keywords

• nonn
• easy

Links

• Clark Kimberling, Table of n, a(n) for n = 1..500
• Index entries for linear recurrences with constant coefficients, signature (5,-8,2,6,-4,-1,1).

Crossrefs

Cf. A213590, A213500.

Programs

GAP

F:=Fibonacci;; List([1..40], n-> n*F(n+7) -2*F(n+9) +2*(n^2+10*n+ 34)); # G. C. Greubel, Jul 06 2019

Magma

F:=Fibonacci; [n*F(n+7) -2*F(n+9) +2*(n^2+10*n+34): n in [1..40]]; // G. C. Greubel, Jul 06 2019

Mathematica

(* First program *)
b[n_]:= n^2; c[n_]:= Fibonacci[n];
T[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
TableForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[T[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213590 *)
r[n_]:= Table[T[n, k], {k, 40}]  (* columns of antidiagonal triangle *)
Table[T[n, n], {n, 1, 40}] (* A213504 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213557 *)
(* Second program *)
With[{F = Fibonacci}, Table[n*F[n+7] -2*F[n+9] +2*(n^2+10*n+34), {n,40}]] (* G. C. Greubel, Jul 06 2019 *)
LinearRecurrence[{5,-8,2,6,-4,-1,1},{1,6,23,70,184,438,971},30] (* Harvey P. Dale, Jun 04 2025 *)

PARI

vector(40, n, f=fibonacci; n*f(n+7) -2*f(n+9) +2*(n^2+10*n+34)) \\ G. C. Greubel, Jul 06 2019

Sage

f=fibonacci; [n*f(n+7) -2*f(n+9) +2*(n^2+10*n+34) for n in (1..40)] # G. C. Greubel, Jul 06 2019

Formula

a(n) = 5*a(n-1) - 8*a(n-2) + 2*a(n-3) + 6*a(n-4) - 4*a(n-5) - a(n-6) + a(n-7).

G.f.: f(x)/g(x), where f(x) = x*(1 + x + x^2 + x^3) and g(x) = (1 - x)^3 (1 - x - x^2)^2.

a(n) = n*Fibonacci(n+7) - 2*Fibonacci(n+9) + 2*n^2 + 20*n + 68. - G. C. Greubel, Jul 06 2019