The On-Line Encyclopedia of Integer Sequences (OEIS)

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A101975

Triangle read by rows: number of Dyck paths of semilength n with k peaks after the first return (0 <= k < n).
(history; published version)

#9 by Michael De Vlieger at Sat Sep 09 22:28:02 EDT 2023

STATUS

proposed

#8 by Alois P. Heinz at Sat Sep 09 20:26:09 EDT 2023

STATUS

editing

#7 by Alois P. Heinz at Sat Sep 09 20:25:24 EDT 2023

DATA

1, 1, 1, 2, 2, 1, 5, 4, 4, 1, 14, 9, 11, 7, 1, 42, 23, 27, 28, 11, 1, 132, 65, 66, 87, 62, 16, 1, 429, 197, 170, 239, 250, 122, 22, 1, 1430, 626, 471, 627, 829, 630, 219, 29, 1, 4862, 2056, 1398, 1656, 2448, 2553, 1419, 366, 37, 1, 16796, 6918, 4381, 4554, 6803, 8813

STATUS

proposed

#6 by Jon E. Schoenfield at Sat Sep 09 19:33:24 EDT 2023

STATUS

editing

#5 by Jon E. Schoenfield at Sat Sep 09 19:33:17 EDT 2023

NAME

Triangle read by rows: number of Dyck paths of semilength n with k peaks after the first return (0<= k <n).

REFERENCES

E. Deutsch, Dyck path enumeration, Discrete Math., 204, 1999, 167-202.

FORMULA

T(n, 0)=c(n-1), T(n, 1) = sum(c(i), i=0..n-2), T(n, k)= sum(c(j)*binomial(n-1-j, k-1)*binomial(n-1-j, k)/(n-1-j), j=0..n-2) for k>=2, where c(i)=binomial(2i, i)/(i+1) (i=0, 1, ...) are the Catalan numbers (A000108).

EXAMPLE

1;

1,1;

2,2,1;

5,4,4,1;

14,9,11,7,1;

42,23,27,28,11,1;

STATUS

approved

#4 by Russ Cox at Fri Mar 30 17:36:00 EDT 2012

AUTHOR

Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 22 2004

Discussion

Fri Mar 30

17:36

OEIS Server: https://oeis.org/edit/global/173

#3 by N. J. A. Sloane at Fri Feb 27 03:00:00 EST 2009

EXAMPLE

T(4,2)=4 because we have UD|(UD)U(UD)D, UD|U(UD)D(UD), UD|U(UD)(UD)D, and

KEYWORD

nonn,tabl,new

#2 by N. J. A. Sloane at Fri Feb 24 03:00:00 EST 2006

FORMULA

T(n,0)=c(n-1), T(n,1) = sum(c(i),i=0..n-2), T(n,k)= sum(c(j)*binomial(n-1-j,k-1)*binomial(n-1-j,k)/(n-1-j),j=0..n-2) for k>=2, where c(i)=binomial(2i,i)/(i+1) (i=0,1,...) are the Catalan numbers (A000108).

KEYWORD

nonn,tabl,new

#1 by N. J. A. Sloane at Sun Feb 20 03:00:00 EST 2005

NAME

DATA

OFFSET

REFERENCES

FORMULA

EXAMPLE

MAPLE

CROSSREFS

KEYWORD

AUTHOR

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