A089107 Square array T(r,j) (r>=1, j>=1) read by antidiagonals, where T(r,j) is the convoluted convolved Fibonacci number G_j^(r) (see the Moree paper).

1, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 3, 5, 5, 0, 1, 3, 7, 9, 8, 0, 1, 4, 10, 17, 19, 13, 0, 1, 4, 13, 25, 37, 34, 21, 0, 1, 5, 16, 38, 64, 77, 65, 34, 0, 1, 5, 20, 51, 102, 146, 158, 115, 55, 0, 1, 6, 24, 70, 154, 259, 331, 314, 210, 89, 0, 1, 6, 28, 89, 222, 418, 626, 710, 611, 368, 144

Offset

1,6

Links

Table of n, a(n) for n=1..78.

P. Moree, Convoluted convolved Fibonacci numbers

Example

Triangle begins:
1
0 1
0 1 2
0 1 2 3
0 1 3 5 5
Array begins:
[1, 1, 2, 3, 5, 8, 13, 21, ...],
[0, 1, 2, 5, 9, 19, 34, 65, ...],
[0, 1, 3, 7, 17, 37, 77, 158, ...],
[0, 1, 3, 10, 25, 64, 146, 331, ...],
[0, 1, 4, 13, 38, 102, 259, 626, ...],
[0, 1, 4, 16, 51, 154, 418, 1098, ...],
[0, 1, 5, 20, 70, 222, 654, 1817, ...],
[0, 1, 5, 24, 89, 309, 967, 2871, ...],
...........

Maple

with(numtheory): m := proc(r, j) d := divisors(r): f := z->1/(1-z-z^2): W := (1/r)*z*sum(mobius(d[i])*f(z^d[i])^(r/d[i]), i=1..nops(d)): Wser := simplify(series(W, z=0, 30)): coeff(Wser, z^j) end: seq(seq(m(n-q+1, q), q=1..n), n=1..17); # for the sequence read by antidiagonals
with(numtheory): f := z->1/(1-z-z^2): m := proc(r, j) d := divisors(r): W := (1/r)*z*sum(mobius(d[i])*f(z^d[i])^(r/d[i]), i=1..nops(d)): Wser := simplify(series(W, z=0, 80)): coeff(Wser, z^j) end: matrix(10, 10, m); # for the square array

Mathematica

rows = 12;
f[z_] := 1/(1 - z - z^2);
W[r_] := W[r] = (z/r)*Sum[MoebiusMu[d]*f[z^d]^(r/d), {d, Divisors[r]}] + O[z]^(rows+1);
A = Table[CoefficientList[W[r], z] // Rest, {r, 1, rows}];
T[r_, j_] := A[[r, j]];
Table[T[r - j + 1, j], {r, 1, rows}, {j, 1, r}] // Flatten (* Jean-François Alcover, Dec 09 2017, from Maple *)

Crossrefs

Sequence in context: A353174 A233292 A108456 * A361756 A364912 A321449

Adjacent sequences: A089104 A089105 A089106 * A089108 A089109 A089110

Keyword

nonn,tabl,easy

Author

N. J. A. Sloane, Dec 05 2003

Extensions

Edited by Emeric Deutsch, Mar 06 2004

Status

approved