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A194549
Triangle read by rows: T(n,k) = Dyson's rank of the k-th partition of n that does not contain 1 as a part, with partitions in lexicographic order.
9
1, 1, 2, 0, 3, 1, 4, -1, 2, 1, 5, 0, 3, 2, 6, -2, 1, 0, 4, 3, 2, 7, -1, 2, 1, 5, 0, 4, 3, 8, -3, 0, -1, 3, 2, 1, 6, 1, 5, 4, 3, 9, -2, 1, 0, 4, -1, 3, 2, 7, 2, 1, 6, 5, 4, 10, -4, -1, -2, 2, 1, 0, 5, 0, 4, 3, 2, 8, -1, 3, 2, 7, 1, 6, 5, 4, 11, -3, 0, -1, 3, -2
OFFSET
1,3
LINKS
FORMULA
a(n) = A141285(n) - A194548(n).
EXAMPLE
Written as a triangle:
1;
1;
2;
0,3;
1,4;
-1,2,1,5;
0,3,2,6;
-2,1,0,4,3,2,7;
-1,2,1,5,0,4,3,8;
-3,0,-1,3,2,1,6,1,5,4,3,9;
-2,1,0,4,-1,3,2,7,2,1,6,5,4,10;
-4,-1,-2,2,1,0,5,0,4,3,2,8,-1,3,2,7,1,6,5,4,11;
MAPLE
T:= proc(n) local b, l;
b:= proc(n, i, t)
if n=0 then l:=l, i-t
elif i>n then
else b(n-i, i, t+1); b(n, i+1, t)
fi
end;
if n<2 then 1 else l:= NULL; b(n, 2, 0); l fi
end:
seq(T(n), n=1..13); # Alois P. Heinz, Dec 20 2011
MATHEMATICA
T[n_] := Module[{b, l}, b[n0_, i_, t_] :=
If[n0 == 0, l = Append[l, i-t],
If[i>n0, , b[n0-i, i, t+1]; b[n0, i+1, t]]];
If[n<2, {1}, l = {}; b[n, 2, 0]; l]];
Table[T[n], {n, 1, 13}] // Flatten (* Jean-François Alcover, Mar 05 2021, after Alois P. Heinz *)
CROSSREFS
The sum of row n is A000041(n-1). Row n has length A187219(n).
Sequence in context: A344616 A316524 A357630 * A063277 A029178 A082375
KEYWORD
sign,tabf,look
AUTHOR
Omar E. Pol, Dec 11 2011
EXTENSIONS
More terms from Alois P. Heinz, Dec 20 2011
STATUS
approved