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%I #58 Oct 31 2021 10:56:06
%S 1,0,1,0,1,1,0,3,1,1,0,8,4,1,1,0,25,11,4,1,1,0,75,39,12,4,1,1,0,245,
%T 124,42,12,4,1,1,0,800,431,138,43,12,4,1,1,0,2700,1470,490,141,43,12,
%U 4,1,1,0,9225,5160,1704,504,142,43,12,4,1,1,0,32065,18160,6088,1763,507,142,43,12,4,1,1
%N Triangle read by rows: the multiset transform of the balanced binary Lyndon words (A022553).
%H Alois P. Heinz, <a href="/A289978/b289978.txt">Rows n = 0..140, flattened</a>
%H K. T. Chen, R. T. Fox and R. C. Lyndon, <a href="https://doi.org/10.2307/1970044">Free differential calculus IV. The quotient groups of the lower central series</a>, Ann. Math. 68 (1) (1958) 81-95.
%H J.-P. Duval, <a href="https://doi.org/10.1016/0196-6774(83)90017-2">Factorizing words over an ordered Alphabet</a>, J. Algorithms 4 (4) (1983) 363.
%H R. J. Mathar, <a href="/A289978/a289978.pdf">A bijection of Dyck Paths and multisets of Balanced Binary Lyndon Words</a> (2021)
%H <a href="/index/Lu#Lyndon">Index entries for sequences related to Lyndon words</a>
%H <a href="/index/Mu#multiplicative_completely">Index entries for triangles generated by the Multiset Transformation</a>
%F G.f.: Product_{j>=1} 1/(1-y*x^j)^A022553(j). - _Alois P. Heinz_, Jul 25 2017
%e The triangle begins in row 0 and column 0 as:
%e 1;
%e 0 1;
%e 0 1 1;
%e 0 3 1 1;
%e 0 8 4 1 1;
%e 0 25 11 4 1 1;
%e 0 75 39 12 4 1 1;
%e 0 245 124 42 12 4 1 1;
%e 0 800 431 138 43 12 4 1 1;
%e 0 2700 1470 490 141 43 12 4 1 1;
%e 0 9225 5160 1704 504 142 43 12 4 1 1;
%e 0 32065 18160 6088 1763 507 142 43 12 4 1 1;
%e 0 112632 64765 21790 6337 1777 508 142 43 12 4 1 1;
%e 0 400023 232347 78845 22798 6396 1780 508 142 43 12 4 1 1;
%e 0 1432613 840285 286652 82941 23047 6410 1781 508 142 43 12 4 1 1;
%p with(numtheory):
%p g:= proc(n) option remember; `if`(n=0, 1, add(
%p mobius(n/d)*binomial(2*d, d), d=divisors(n))/(2*n))
%p end:
%p b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
%p `if`(min(i, p)<1, 0, add(binomial(g(i)+j-1, j)*
%p b(n-i*j, i-1, p-j), j=0..min(n/i, p)))))
%p end:
%p T:= (n, k)-> b(n$2, k):
%p seq(seq(T(n, k), k=0..n), n=0..14); # _Alois P. Heinz_, Jul 25 2017
%t g[n_]:=g[n]=If[n==0, 1, Sum[MoebiusMu[n/d] Binomial[2d, d], {d, Divisors[n]}]/(2n)]; b[n_, i_, p_]:=b[n, i, p]=If[p>n, 0, If[n==0, 1, If[Min[i, p]<1, 0, Sum[Binomial[g[i] + j - 1, j] b[n - i*j, i - 1, p - j], {j, 0, Min[n/i, p]}]]]]; Table[b[n, n, k], {n, 0, 14}, {k, 0, n}]//Flatten (* _Indranil Ghosh_, Aug 05 2017, after Maple code *)
%t nn = 14;
%t b[n_] := If[n==0, 1, Sum[MoebiusMu[n/d] Binomial[2d, d], {d, Divisors[n]}]/ (2n)];
%t CoefficientList[#, y]& /@ (Series[Product[1/(1 - y x^i)^b[i], {i, 1, nn}], {x, 0, nn}] // CoefficientList[#, x]&) // Flatten (* _Jean-François Alcover_, Oct 29 2021 *)
%Y Cf. A022553 (column k=1), A000108 (row sums), A033184, A290277.
%Y T(2n,n) gives A292287.
%K nonn,tabl
%O 0,8
%A _R. J. Mathar_, Jul 18 2017
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