|
|
A032021
|
|
Number of compositions (ordered partitions) of n into distinct odd parts.
|
|
26
|
|
|
1, 1, 0, 1, 2, 1, 2, 1, 4, 7, 4, 7, 6, 13, 6, 19, 32, 25, 32, 31, 58, 43, 82, 49, 132, 181, 156, 193, 230, 325, 278, 457, 376, 715, 448, 967, 1290, 1345, 1386, 1723, 2276, 2341, 3116, 2959, 4750, 3823, 6358, 4681, 9480, 10945, 11832, 12169, 16442, 18793, 21002, 25537, 27820, 37687
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,5
|
|
LINKS
|
|
|
FORMULA
|
"AGK" (ordered, elements, unlabeled) transform of 1, 0, 1, 0...(odds)
G.f.: sum(k>=0, k! * x^(k^2) / prod(j=1..k, 1-x^(2*j) ) ). - Vladeta Jovovic, Aug 05 2004
|
|
MAPLE
|
b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)
->x+y, b(n, i-2), [0, `if`(i>n, [], b(n-i, i-2))[]], 0)))
end:
a:= proc(n) local l; l:= b(n, n-1+irem(n, 2));
a(n):= add(l[i]*(i-1)!, i=1..nops(l))
end:
|
|
MATHEMATICA
|
b[n_, i_] := If[n == 0, {1}, If[i<1, {}, Plus @@ PadRight[{b[n, i-2], Join[{0}, If[i>n, {}, b[n-i, i-2]]]}]]]; a[n_] := Module[{l}, l = b[n, n-1+Mod[n, 2]]; Sum[l[[i]]*(i-1)!, {i, 1, Length[l]}]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jan 30 2014, after Alois P. Heinz *)
|
|
PROG
|
(PARI)
N=66; q='q+O('q^N);
gf=sum(k=0, N, k! * q^(k^2) / prod(j=1, k, 1-q^(2*j) ) );
Vec(gf)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|