A213579 Rectangular array: (row n) = b**c, where b(h) = F(h), c(h) = n-1+h, where F=A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.

1, 3, 2, 7, 5, 3, 14, 11, 7, 4, 26, 21, 15, 9, 5, 46, 38, 28, 19, 11, 6, 79, 66, 50, 35, 23, 13, 7, 133, 112, 86, 62, 42, 27, 15, 8, 221, 187, 145, 106, 74, 49, 31, 17, 9, 364, 309, 241, 178, 126, 86, 56, 35, 19, 10, 596, 507, 397, 295, 211, 146, 98, 63, 39, 21

Offset: 1

Author

Clark Kimberling, Jun 18 2012

Keywords

• nonn
• tabl
• easy

Comments

Principal diagonal: A213580.

Antidiagonal sums: A053808.

Row 1, (1,1,2,3,5,...)**(1,2,3,4,...): A001924.

Row 2, (1,1,2,3,5,...)**(2,3,4,5,...): A023548.

Row 3, (1,1,2,3,5,...)**(3,4,5,6,...): A023552.

Row 4, (1,1,2,3,5,...)**(4,5,6,7,...): A210730.

Row 5, (1,1,2,3,5,...)**(5,6,7,8,...): A210731.

For a guide to related arrays, see A213500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1....3....7....14...26...46
2....5....11...21...38...66
3....7....15...28...50...86
4....9....19...35...62...106
5....11...23...42...74...126
6....13...27...49...86...146
		

Links

• Clark Kimberling, Antidiagonas n = 1..60, flattened

Crossrefs

Cf. A000045, A213500.

Programs

GAP

Flat(List([1..12], n-> List([1..n], k-> Fibonacci(k+3) + n*Fibonacci(k+2) -(n+k+2) ))); # G. C. Greubel, Jul 08 2019

Magma

[[Fibonacci(k+3) + n*Fibonacci(k+2) -(n+k+2): k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jul 08 2019

Mathematica

(* First program *)
b[n_]:= Fibonacci[n]; c[n_]:= 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}]] (* A213579 *)
r[n_]:= Table[T[n, k], {k, 40}]
d = Table[T[n, n], {n, 1, 40}] (* A213580 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A053808 *)
(* Second program *)
Table[Fibonacci[n-k+4] +k*Fibonacci[n-k+3] -(n+3), {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 08 2019 *)

PARI

t(n,k) = fibonacci(n-k+4) + k*fibonacci(n-k+3) - (n+3);
for(n=1,12, for(k=1,n, print1(t(n,k), ", "))) \\ G. C. Greubel, Jul 08 2019

Sage

[[fibonacci(k+3) + n*fibonacci(k+2) -(n+k+2) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 08 2019

Formula

T(n,k) = 3*T(n,k-1) - 2*T(n,k-2) - T(n,k-3) + T(n,k-4).

G.f. for row n: f(x)/g(x), where f(x) = n - (n-1)*x and g(x) = (1-x-x^2) *(1-x)^2.

T(n, k) = Fibonacci(k+3) + n*Fibonacci(k+2) - (n+k+2). - G. C. Greubel, Jul 08 2019