A372873 - OEIS

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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, 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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

Table of n, a(n) for n=1..66.

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:

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

Cf. A000007, A000079, A001788, A007051 (row sums), A033322, A372874.

Sequence in context: A099096 A099089 A121298 * A212206 A247489 A208756

Adjacent sequences: A372870 A372871 A372872 * A372874 A372875 A372876

KEYWORD

nonn,tabl

AUTHOR

Stefano Spezia, May 15 2024

STATUS

approved