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A110616 A convolution triangle of numbers based on A001764. 5
1, 1, 1, 3, 2, 1, 12, 7, 3, 1, 55, 30, 12, 4, 1, 273, 143, 55, 18, 5, 1, 1428, 728, 273, 88, 25, 6, 1, 7752, 3876, 1428, 455, 130, 33, 7, 1, 43263, 21318, 7752, 2448, 700, 182, 42, 8, 1, 246675, 120175, 43263, 13566, 3876, 1020, 245, 52, 9, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Reflected version of A069269. - Vladeta Jovovic, Sep 27 2006

With offset 1 for n and k, T(n,k) = number of Dyck paths of semilength n for which all descents are of even length (counted by A001764) with no valley vertices at height 1 and with k returns to ground level. For example, T(3,2)=2 counts U^4 D^4 U^2 D^2, U^2 D^2 U^4 D^4 where U=upstep, D=downstep and exponents denote repetition. - David Callan, Aug 27 2009

Riordan array (f(x), x*f(x)) with f(x) = (2/sqrt(3*x))*sin((1/3)*arcsin(sqrt(27*x/4))). - Philippe Deléham, Jan 27 2014

Antidiagonals of convolution matrix of Table 1.4, p. 397, of Hoggatt and Bicknell. - Tom Copeland, Dec 25 2019

LINKS

Table of n, a(n) for n=0..54.

Naiomi Cameron, J. E. McLeod, Returns and Hills on Generalized Dyck Paths, Journal of Integer Sequences, Vol. 19, 2016, #16.6.1.

V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395-405.

Sheng-Liang Yang, LJ Wang, Taylor expansions for the m-Catalan numbers, Australasian Journal of Combinatorics, Volume 64(3) (2016), Pages 420-431.

FORMULA

T(n, k) = Sum_{j>=0} T(n-1, k-1+j)*A000108(j); T(0, 0) = 1; T(n, k) = 0 if k < 0 or if k > n.

G.f.: 1/(1 - x*y*TernaryGF) = 1 + (y)x + (y+y^2)x^2 + (3y+2y^2+y^3)x^3 +... where TernaryGF = 1 + x + 3x^2 + 12x^3 + ... is the GF for A001764. - David Callan, Aug 27 2009

T(n, k) = ((k+1)*binomial(3*n-2*k,2*n-k))/(2*n-k+1). - Vladimir Kruchinin, Nov 01 2011

EXAMPLE

Triangle begins:

1;

1, 1;

3, 2, 1;

12, 7, 3, 1;

55, 30, 12, 4, 1;

273, 143, 55, 18, 5, 1;

1428, 728, 273, 88, 25, 6, 1;

7752, 3876, 1428, 455, 130, 33, 7, 1;

43263, 21318, 7752, 2448, 700, 182, 42, 8, 1;

246675, 120175, 43263, 13566, 3876, 1020, 245, 52, 9, 1;

...

MATHEMATICA

Table[(k + 1) Binomial[3 n - 2 k, 2 n - k]/(2 n - k + 1), {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jun 28 2017 *)

PROG

(Maxima) T(n, k):=((k+1)*binomial(3*n-2*k, 2*n-k))/(2*n-k+1); // Vladimir Kruchinin, Nov 01 2011

CROSSREFS

Successive columns: A001764, A006013, A001764, A006629, A102893, A006630, A102594, A006631; row sums: A098746; see also A092276.

Sequence in context: A184182 A118435 A115085 * A059418 A092582 A213262

Adjacent sequences: A110613 A110614 A110615 * A110617 A110618 A110619

KEYWORD

nonn,tabl

AUTHOR

Philippe Deléham, Sep 14 2005, Jun 15 2007

STATUS

approved

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Last modified December 1 23:44 EST 2022. Contains 358485 sequences. (Running on oeis4.)