A223968
Square array T, read by antidiagonals: T(n,k) = 0 if n-k >= 5 or if k-n >= 6, T(4,0) = T(3,0) = T(2,0) = T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = T(0,5) = 1, T(n,k) = T(n-1,k) + T(n,k-1).
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 0, 0, 6, 15, 20, 15, 5, 0, 0, 6, 21, 35, 35, 20, 0, 0, 0, 0, 27, 56, 70, 55, 20, 0, 0, 0, 0, 27, 83, 126, 125, 75, 0, 0, 0, 0, 0, 0, 110, 209, 251, 200, 75, 0, 0, 0, 0, 0, 0, 110, 319, 460, 451, 275, 0, 0, 0, 0, 0, 0, 0
Offset: 0
Square array begins:
1....1....1....1....1....1....0....0....0....0....0....0
1....2....3....4....5....6....6....0....0....0....0....0
1....3....6...10...15...21...27...27....0....0....0....0
1....4...10...20...35...56...83..110..110....0....0....0
1....5...15...35...70..126..209..319..429..429....0....0
0....5...20...55..125..251..460..779.1208.1637.1637....0
0....0...20...75..200..451..911.1690.2898.4535.6172.6172
...
Square array, read by diagonals, with 0 omitted:
1, 5, 20, 75, 275, 1001, 3639, 13243, 48280, ...
1, 5, 20, 75, 275, 1001, 3639, 13243, 48280, ...
1, 4, 15, 55, 200, 726, 2638, 9604, 35037, ...
1, 3, 10, 35, 125, 451, 1637, 5965, 21794, ...
1, 2, 6, 20, 70, 251, 911, 3327, 12190, 44744, ...
1, 3, 10, 35, 126, 460, 1690, 6225, 22950, ...
1, 4, 15, 56, 209, 779, 2898, 10760, 39882, ...
1, 5, 21, 83, 319, 1208, 4535, 16932, 62986, ...
1, 6, 27, 110, 429, 1637, 6172, 23104, 86090, ...
1, 6, 27, 110, 429, 1637, 6172, 23104, 86090, ...
Cf. Similar sequences:
A214846,
A216054,
A216201,
A216210,
A216216,
A216218,
A216219,
A216220,
A216226,
A216228,
A216229,
A216230,
A216232,
A216235,
A216236,
A216238,
A217257,
A217315,
A217593,
A217765,
A217770
A336675
Number of paths of length n starting at initial node of the path graph P_10.
Original entry on oeis.org
1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 251, 460, 911, 1690, 3327, 6225, 12190, 22950, 44744, 84626, 164407, 312019, 604487, 1150208, 2223504, 4239225, 8181175, 15621426, 30108147, 57556155, 110820165, 212037241, 407946421, 781074572, 1501844193, 2877011660, 5529362694
Offset: 0
- Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015-2016.
- Nachum Dershowitz, Between Broadway and the Hudson: A Bijection of Corridor Paths, arXiv:2006.06516 [math.CO], 2020.
- Index entries for linear recurrences with constant coefficients, signature (1,4,-3,-3,1).
Cf.
A000004 (row 0),
A000007 (row 1),
A000012 (row 2),
A016116 (row 3),
A000045 (row 4),
A038754 (row 5),
A028495 (row 6),
A030436 (row 7),
A061551 (row 8),
A178381 (row 9), this sequence (row 10),
A336678 (row 11),
A001405 (limit).
-
X := j -> (-1)^(j/11) - (-1)^(1-j/11):
a := k -> add((2 + X(j))*X(j)^k, j in [1, 3, 5, 7, 9])/11:
seq(simplify(a(n)), n=0..30); # Peter Luschny, Sep 17 2020
-
a[n_,m_]:=2^(n+1)/(m+1) Module[{x=(Pi r)/(m+1)},Sum[Cos[x]^n (1+Cos[x]),{r,1,m,2}]]
Table[a[n,10],{n,0,40}]//Round (* Herbert Kociemba, Sep 14 2020 *)
-
my(x='x+O('x^44)); Vec((1 - 3*x^2 + x^4)/(1 - x - 4*x^2 + 3*x^3 + 3*x^4 - x^5)) \\ Joerg Arndt, Jul 31 2020
Showing 1-2 of 2 results.
Comments