A089112 - OEIS

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).

%I #15 Mar 27 2018 08:50:58

%S 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,

%T 13,25,37,37,21,0,1,5,16,38,64,77,65,34,0,1,5,20,54,102,146,158,120,

%U 55,0,1,6,24,70,154,259,331,314,210,89,0,1,7,28,89,222,425,626,710,611,376,144

%N 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).

%H P. Moree, <a href="https://arxiv.org/abs/math/0311205">Convoluted convolved Fibonacci numbers</a>, arXiv:math/0311205 [math.CO], 2003.

%H P. Moree, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL7/Moree/moree12.htm">Convoluted Convolved Fibonacci Numbers</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.2.

%e Triangle begins:

%e 1

%e 1 1

%e 0 1 2

%e 0 1 3 3

%e 0 1 3 5 5

%e Array begins:

%e 1, 1, 2, 3, 5, 8, 13, 21, ...,

%e 1, 1, 3, 5, 11, 19, 37, 65, ...,

%e 0, 1, 3, 7, 17, 37, 77, 158, ...,

%e 0, 1, 3, 10, 25, 64, 146, 331, ...,

%e 0, 1, 4, 13, 38, 102, 259, 626, ...,

%e 0, 1, 5, 16, 54, 154, 425, 1098, ...,

%e 0, 1, 5, 20, 70, 222, 654, 1817, ...,

%e 0, 1, 5, 24, 89, 309, 967, 2871, ...,

%e ...

%p 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

%p 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

%t 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}] &;

%t Table[m[r - j + 1, j], {r, 1, 12}, {j, 1, r}] // Flatten (* _Jean-François Alcover_, Mar 25 2018, translated from Maple *)

%K nonn,tabl,easy

%O 1,6

%A _N. J. A. Sloane_, Dec 05 2003

%E Edited by _Emeric Deutsch_, Mar 06 2004