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A134426 Triangle read by rows: T(n,k) is the number of paths of length n in the first quadrant, starting at the origin, ending at height k and consisting of 2 kind of upsteps U=(1,1) (U1 and U2), 3 kind of flatsteps F=(1,0) (F1, F2 and F3) and 1 kind of downsteps D=(1,-1). 1
1, 3, 2, 11, 12, 4, 45, 62, 36, 8, 197, 312, 240, 96, 16, 903, 1570, 1440, 784, 240, 32, 4279, 7956, 8244, 5472, 2320, 576, 64, 20793, 40670, 46116, 35224, 18480, 6432, 1344, 128, 103049, 209712, 254912, 216384, 132320, 57600, 17024, 3072, 256 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

T(n,0)=A001003(n+1) (the little Schroeder numbers). Row sums yield A134425.

LINKS

Table of n, a(n) for n=0..44.

FORMULA

T(n,k)=[(k+1)2^k/(n+1)]Sum(binom(n+1,j)binom(n+1,k+j+1)2^j, j=0..n-k) (0<=k<=n). G.f.=g/(1-2tzg), where g=1+3zg+2z^2 g^2 is the g.f. of the little Schroeder numbers 1,3,11,45,197,... (A001003).

EXAMPLE

T(2,1)=12 because we have 6 paths of shape FU and 6 paths of shape UF.

Triangle starts:

1;

3,2;

11,12,4;

45,62,36,8;

197,312,240,96,16;

MAPLE

T:=proc(n, k) options operator, arrow: 2^k*(k+1)*(sum(2^j*binomial(n+1, j)*binomial(n+1, k+1+j), j=0..n-k))/(n+1) end proc: for n from 0 to 8 do seq(T(n, k), k=0..n) end do; # yields sequence in triangular form

CROSSREFS

Cf. A001003, A134425.

Sequence in context: A258386 A159610 A074246 * A122672 A194608 A297870

Adjacent sequences:  A134423 A134424 A134425 * A134427 A134428 A134429

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Nov 05 2007

STATUS

approved

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Last modified March 26 16:55 EDT 2019. Contains 321510 sequences. (Running on oeis4.)