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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]

T(n, k) = Sum_{j, 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]

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;

...

...

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

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GF: 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]

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

V. E. Hoggatt, Jr. and M. Bicknell, <a href="http://www.fq.math.ca/Scanned/14-5/hoggatt1.pdf">Catalan and related sequences arising from inverses of Pascal's triangle matrices</a>, Fib. Quart., 14 (1976), 395-405.

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