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A372873
Triangle read by rows: T(n,k) is the number of flattened Catalan words of length n with exactly k runs of descents.
1
1, 0, 2, 0, 1, 4, 0, 0, 6, 8, 0, 0, 1, 24, 16, 0, 0, 0, 10, 80, 32, 0, 0, 0, 1, 60, 240, 64, 0, 0, 0, 0, 14, 280, 672, 128, 0, 0, 0, 0, 1, 112, 1120, 1792, 256, 0, 0, 0, 0, 0, 18, 672, 4032, 4608, 512, 0, 0, 0, 0, 0, 1, 180, 3360, 13440, 11520, 1024
OFFSET
1,3
LINKS
Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See p. 11.
FORMULA
G.f.: x*y*(1 - 2*x*y)/(1 - 4*x*y - x^2*y + 4*x^2*y^2).
T(n,k) = 2^(2*k-n-1)*binomial(n-1, 2*(n-k)).
T(n,n) = A000079(n-1).
T(n,n-1) = A001788(n-2).
T(n,1) = A000007(n-1).
T(n,2) = A033322(n-1).
Sum_{k>=0} T(n,k) = A007051(n-1).
EXAMPLE
The triangle begins:
1;
0, 2;
0, 1, 4;
0, 0, 6, 8;
0, 0, 1, 24, 16;
0, 0, 0, 10, 80, 32;
0, 0, 0, 1, 60, 240, 64;
0, 0, 0, 0, 14, 280, 672, 128;
0, 0, 0, 0, 1, 112, 1120, 1792, 256;
...
T(4,3) = 6 since there 6 flattened Catalan words of length 4 with 3 runs of descents: 0010, 0100, 0101, 0110, 0120, and 0121.
MATHEMATICA
T[n_, k_]:=SeriesCoefficient[x*y*(1-2*x*y)/(1-4*x*y-x^2*y+4x^2*y^2), {x, 0, n}, {y, 0, k}]; Table[T[n, k], {n, 11}, {k, n}]//Flatten (* or *)
T[n_, k_]:=2^(2*k-n-1)*Binomial[n-1, 2*(n-k)]; Table[T[n, k], {n, 11}, {k, n}]//Flatten
CROSSREFS
Sequence in context: A099096 A099089 A121298 * A212206 A247489 A208756
KEYWORD
nonn,tabl
AUTHOR
Stefano Spezia, May 15 2024
STATUS
approved