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A238125 Triangle read by rows: T(n,k) gives the number of ballot sequences of length n having exactly k flat steps, n>=0, 0<=k<=n. 13
1, 1, 0, 1, 1, 0, 2, 1, 1, 0, 4, 3, 2, 1, 0, 9, 8, 6, 2, 1, 0, 22, 24, 17, 9, 3, 1, 0, 59, 70, 57, 29, 13, 3, 1, 0, 170, 224, 191, 108, 49, 17, 4, 1, 0, 516, 744, 663, 399, 201, 69, 23, 4, 1, 0, 1658, 2588, 2415, 1573, 802, 322, 104, 28, 5, 1, 0, 5583, 9317, 9108, 6249, 3343, 1408, 510, 137, 35, 5, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,7
COMMENTS
Also number of standard Young tableaux with n cells and exactly k successions. A succession is a pair of cells (v, v+1) lying in the same row.
T(2n,n) gives A241785.
Row sums are A000085.
LINKS
Joerg Arndt and Alois P. Heinz, Rows n = 0..45, flattened
EXAMPLE
Triangle starts:
00: 1;
01: 1, 0;
02: 1, 1, 0;
03: 2, 1, 1, 0;
04: 4, 3, 2, 1, 0;
05: 9, 8, 6, 2, 1, 0;
06: 22, 24, 17, 9, 3, 1, 0;
07: 59, 70, 57, 29, 13, 3, 1, 0;
08: 170, 224, 191, 108, 49, 17, 4, 1, 0;
09: 516, 744, 663, 399, 201, 69, 23, 4, 1, 0;
10: 1658, 2588, 2415, 1573, 802, 322, 104, 28, 5, 1, 0;
11: 5583, 9317, 9108, 6249, 3343, 1408, 510, 137, 35, 5, 1, 0;
12: 19683, 34924, 35695, 25642, 14368, 6440, 2411, 751, 189, 42, 6, 1, 0;
...
MAPLE
b:= proc(n, v, l) option remember; `if`(n<1, 1, expand(
add(`if`(i=1 or l[i-1]>l[i], `if`(i=v, x, 1)*
b(n-1, i, subsop(i=l[i]+1, l)), 0), i=1..nops(l))+
b(n-1, nops(l)+1, [l[], 1])))
end:
T:= n-> seq(coeff(b(n-1, 1, [1]), x, i), i=0..n):
seq(T(n), n=0..12);
MATHEMATICA
b[n_, v_, l_List] := b[n, v, l] = If[n<1, 1, Sum[If[i == 1 || l[[i-1]] > l[[i]], If[i == v, x, 1]*b[n-1, i, ReplacePart[l, i -> l[[i]]+1]], 0], {i, 1, Length[l]}] + b[n-1, Length[l]+1, Append[l, 1]]]; T[n_] := Table[Coefficient[b[n-1, 1, {1}], x, i], {i, 0, n}]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 07 2015, translated from Maple *)
CROSSREFS
Sequence in context: A255704 A191347 A106234 * A062507 A238750 A131044
KEYWORD
nonn,tabl
AUTHOR
Joerg Arndt and Alois P. Heinz, Feb 21 2014
STATUS
approved

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Last modified March 19 02:51 EDT 2024. Contains 370952 sequences. (Running on oeis4.)