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%I #9 Dec 09 2017 11:08:05
%S 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,
%T 25,37,34,21,0,1,5,16,38,64,77,65,34,0,1,5,20,51,102,146,158,115,55,0,
%U 1,6,24,70,154,259,331,314,210,89,0,1,6,28,89,222,418,626,710,611,368,144
%N 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).
%H P. Moree, <a href="http://arXiv.org/abs/math.CO/0311205">Convoluted convolved Fibonacci numbers</a>
%e Triangle begins:
%e 1
%e 0 1
%e 0 1 2
%e 0 1 2 3
%e 0 1 3 5 5
%e Array begins:
%e [1, 1, 2, 3, 5, 8, 13, 21, ...],
%e [0, 1, 2, 5, 9, 19, 34, 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, 4, 16, 51, 154, 418, 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)): 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): 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
%t rows = 12;
%t f[z_] := 1/(1 - z - z^2);
%t W[r_] := W[r] = (z/r)*Sum[MoebiusMu[d]*f[z^d]^(r/d), {d, Divisors[r]}] + O[z]^(rows+1);
%t A = Table[CoefficientList[W[r], z] // Rest, {r, 1, rows}];
%t T[r_, j_] := A[[r, j]];
%t Table[T[r - j + 1, j], {r, 1, rows}, {j, 1, r}] // Flatten (* _Jean-François Alcover_, Dec 09 2017, 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
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