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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). 0
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Dec 09 2017, from Maple *)

CROSSREFS

Sequence in context: A289251 A233292 A108456 * A321449 A180279 A179968

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

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Last modified January 18 05:18 EST 2021. Contains 340250 sequences. (Running on oeis4.)