A089112 Square array T(r,j) (r >= 1, j >= 1) read by antidiagonals, where T(r,j) is the sign twisted convoluted convolved Fibonacci number H_j^(r) (see the Moree paper).

1, 1, 1, 0, 1, 2, 0, 1, 3, 3, 0, 1, 3, 5, 5, 0, 1, 3, 7, 11, 8, 0, 1, 4, 10, 17, 19, 13, 0, 1, 5, 13, 25, 37, 37, 21, 0, 1, 5, 16, 38, 64, 77, 65, 34, 0, 1, 5, 20, 54, 102, 146, 158, 120, 55, 0, 1, 6, 24, 70, 154, 259, 331, 314, 210, 89, 0, 1, 7, 28, 89, 222, 425, 626, 710, 611, 376, 144

Offset

1,6

Links

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

P. Moree, Convoluted convolved Fibonacci numbers, arXiv:math/0311205 [math.CO], 2003.

P. Moree, Convoluted Convolved Fibonacci Numbers, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.2.

Example

Triangle begins:
  1
  1 1
  0 1 2
  0 1 3 3
  0 1 3 5 5
Array begins:
  1, 1, 2,  3,  5,   8,  13,   21, ...,
  1, 1, 3,  5, 11,  19,  37,   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, 5, 16, 54, 154, 425, 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)): (-1)^r*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): 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, 80)): (-1)^r*coeff(Wser, z^j) end: matrix(10, 10, m); # for the square array

Mathematica

f[z_] = -1/(1-z-z^2); m[r_, j_] := (-1)^r *(1/r)*z*DivisorSum[r, MoebiusMu[#] * f[z^#]^(r/#) &] // SeriesCoefficient[#, {z, 0, j}] &;
Table[m[r - j + 1, j], {r, 1, 12}, {j, 1, r}] // Flatten (* Jean-François Alcover, Mar 25 2018, translated from Maple *)

Crossrefs

Sequence in context: A331103 A267724 A179329 * A155584 A139600 A198321

Adjacent sequences: A089109 A089110 A089111 * A089113 A089114 A089115

Keyword

nonn,tabl,easy

Author

N. J. A. Sloane, Dec 05 2003

Extensions

Edited by Emeric Deutsch, Mar 06 2004

Status

approved