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A145598
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Triangular array of generalised Narayana numbers: T(n,k) = 4/(n+1)*binomial(n+1,k+3)*binomial(n+1,k-1).
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5
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1, 4, 4, 10, 24, 10, 20, 84, 84, 20, 35, 224, 392, 224, 35, 56, 504, 1344, 1344, 504, 56, 84, 1008, 3780, 5760, 3780, 1008, 84, 120, 1848, 9240, 19800, 19800, 9240, 1848, 120, 165, 3168, 20328, 58080, 81675, 58080, 20328, 3168, 165, 220, 5148, 41184, 151008
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 3,2
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COMMENTS
| T(n,k) is the number of walks of n unit steps, each step in the direction either up (U), down (D), right (R) or left (L), starting from (0,0) and finishing at lattice points on the horizontal line y = 3 and which remain in the upper half-plane y >= 0. An example is given in the Example section below.
The current array is the case r = 3 of the generalised Narayana numbers N_r(n,k) := (r + 1)/(n + 1)*binomial(n + 1,k + r)*binomial(n + 1,k - 1), which count walks of n steps from the origin to points on the horizontal line y = r that remain in the upper half-plane. Case r = 0 gives the table of Narayana numbers A001263 (but with an offset of 0 in the row numbering). For other cases see A145596 (r = 1), A145597 (r = 2) and A145599 (r = 4).
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LINKS
| R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6
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FORMULA
| T(n,k) = 4/(n+1)*binomial(n+1,k+3)*binomial(n+1,k-1) for n >=3 and 1 <= k <= n-2. In the notation of [Guy], T(n,k) equals w_n(x,y) at (x,y) = (2*k - n + 1,3). Row sums A003518.
O.g.f. for column k+2: 4/(k + 1) * y^(k+4)/(1 - y)^(k+6) * Jacobi_P(k,4,1,(1 + y)/(1 - y)).
Identities for row polynomials R_n(x) := sum {k = 1 .. n - 2} T(n,k)*x^k:
x^3*R_(n-1)(x) = 4*(n - 1)*(n - 2)*(n - 3)/((n + 1)*(n + 2)*(n + 3)*(n + 4)) * sum {k = 0..n} binomial(n + 4,k) * binomial(2n - k,n) * (x - 1)^k;
sum {k = 1..n} (-1)^(k+1)*binomial(n,k)*R_k(x^2)*(1 + x)^(2*(n-k)) = R_n(1)*x^(n-1) = A003518(n)*x^(n-1).
Row generating polynomial R_(n+3)(x) = 4/(n+4)*x*(1-x)^n * Jacobi_P(n,4,4,(1+x)/(1-x)). [From Peter Bala (pbala(AT)toucansurf.com), Oct 31 2008]
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EXAMPLE
| Triangle starts
n\k|..1.....2....3.....4.....5.....6
====================================
.3.|..1
.4.|..4.....4
.5.|.10....24...10
.6.|.20....84...84....20
.7.|.35...224..392...224....35
.8.|.56...504.1344..1344...504....56
...
Row 5:
T(5,3) = 10: the 10 walks of length 5 from (0,0) to (2,3) are
UUURR, UURUR, UURRU, URUUR, URURU, URRUU, RUUUR, RUURU, RURUU
and RRUUU.
*
*......*......*......y......*......*......*
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*.....10......*.....24......*.....10......*
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*......*......*......*......*......*......*
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*......*......*......*......*......*......*
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*......*......*......o......*......*......* x axis
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MAPLE
| with(combinat):
T:= (n, k) -> 4/(n+1)*binomial(n+1, k+3)*binomial(n+1, k-1):
for n from 3 to 12 do
seq(T(n, k), k = 1 .. n-2);
end do;
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CROSSREFS
| A003518 (row sums), A001263, A145602, A145596, A145597, A145599.
Sequence in context: A185779 A095009 A178820 * A117881 A161433 A180498
Adjacent sequences: A145595 A145596 A145597 * A145599 A145600 A145601
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Peter Bala (pbala(AT)toucansurf.com), Oct 15 2008
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