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A035317
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Pascal-like triangle associated with A000670.
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17
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1, 1, 1, 1, 2, 2, 1, 3, 4, 2, 1, 4, 7, 6, 3, 1, 5, 11, 13, 9, 3, 1, 6, 16, 24, 22, 12, 4, 1, 7, 22, 40, 46, 34, 16, 4, 1, 8, 29, 62, 86, 80, 50, 20, 5, 1, 9, 37, 91, 148, 166, 130, 70, 25, 5, 1, 10, 46, 128, 239, 314, 296, 200, 95, 30, 6, 1, 11, 56, 174, 367, 553, 610, 496, 295, 125
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| From Johannes W. Meijer, Jul 20 2011: (Start)
The triangle sums, see A180662 for their definitions, link this ‘Races with Ties’ triangle with several sequences, see the crossrefs. Observe that the Kn4 sums lead to the golden rectangle numbers A001654 and that the Fi1 and Fi2 sums lead to the Jacobsthal sequence A001045.
The series expansion of G(x, y) = 1/((y*x-1)*(y*x+1)*((y+1)*x-1)) as function of x leads to this sequence, see the second Maple program. (End)
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REFERENCES
| E. Mendelson, Races with Ties, Math. Mag. 55 (1982), 170-175.
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FORMULA
| T(n, k)= Sum {j=0..floor(n/2) binomial(n-2j, k-2j)} - Paul Barry (pbarry(AT)wit.ie), Feb 11 2003
From Johannes W. Meijer, Jul 20 2011: (Start)
T(n, k) = sum((-1)^(i+k) * binomial(i+n-k+1,i), i=0..k) (Mendelson)
T(n, k) = T(n-1, k-1) + T(n-1, k) with T(n, 0) = 1 and T(n, n) = floor(n/2) + 1 (Mendelson)
sum((-1)^k * (n-k+1)^n * T(n, k), k = 0..n) = A000670(n) (Mendelson)
T(n, n-k) = A128176(n, k); T(n+k, n-k) = A158909(n, k); T(2*n-k, k) = A092879(n, k) (End)
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EXAMPLE
| Triangle begins:
1;
1,1;
1,2,2;
1,3,4,2;
1,4,7,6,3;
1,5,11,13,9,3;
1,6,16,24,22,12,4;
1,7,22,40,46,34,16,4;
1,8,29,62,86,80,50,20,5;
1,9,37,91,148,166,130,70,25,5;
1,10,46,128,239,314,296,200,95,30,6;
...
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MAPLE
| From Johannes W. Meijer, Jul 20 2011: (Start)
A035317 := proc(n, k): add((-1)^(i+k) * binomial(i+n-k+1, i), i=0..k) end: seq(seq(A035317(n, k), k=0..n), n=0..10);
A035317 := proc(n, k): coeff(coeftayl(1/((y*x-1)*(y*x+1)*((y+1)*x-1)), x=0, n), y, k) end: seq(seq(A035317(n, k), k=0..n), n=0..10); (End)
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MATHEMATICA
| t[n_, k_] := (-1)^k*(((-1)^k*(n+2)!*Hypergeometric2F1[1, n+3, k+2, -1])/((k+1)!*(n-k+1)!) + 2^(k-n-2)); Flatten[ Table[ t[n, k], {n, 0, 11}, {k, 0, n}]] (* From Jean-François Alcover, Dec 14 2011, after Johannes W. Meijer *)
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CROSSREFS
| Row sums are A000975, diagonal sums are A080239.
Similar to the triangles A059259, A080242, A108561, A112555.
Cf. A059260.
Triangle sums (see the comments): A000975 (Row1), A059841 (Row2), A080239 (Kn11), A052952 (Kn21), A129696 (Kn22), A001906 (Kn3), A001654 (Kn4), A001045 (Fi1, Fi2), A023435 (Ca2), Gi2 (A193146), A190525 (Ze2), A193147 (Ze3), A181532 (Ze4). [Johannes W. Meijer, Jul 20 2011]
Sequence in context: A071453 A080242 A183927 * A103923 A186711 A061987
Adjacent sequences: A035314 A035315 A035316 * A035318 A035319 A035320
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KEYWORD
| nonn,easy,tabl,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu)
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