Original entry on oeis.org
1, 1, 2, 4, 9, 20, 45, 100, 220, 478, 1025, 2169, 4534, 9364, 19125, 38646, 77306, 153173, 300765, 585518, 1130612, 2166284, 4120062, 7780817, 14595364, 27201794, 50383690, 92768457, 169835952, 309223286, 560036477, 1009124256
Offset: 0
- Alter, Ronald; Curtz, Thaddeus B.; Wang, Chung C. Permutations with fixed index and number of inversions. Proceedings of the Fifth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1974), pp. 209-228. Congressus Numerantium, No. X, Utilitas Math., Winnipeg, Man., 1974.
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nmax = 20; p = 1; Do[Do[p = Expand[p*(1 - x^i*y^j)]; p = Select[p, (Exponent[#, x] <= nmax) && (Exponent[#, y] <= nmax) &], {i, 1, nmax}], {j, 1, nmax}]; p = Expand[Normal[Series[1/p, {x, 0, nmax}, {y, 0, nmax}]]]; p = Select[p, Exponent[#, x] == Exponent[#, y] &]; Flatten[{1, Table[Coefficient[p, x^n*y^n], {n, 1, nmax}]}] (* Vaclav Kotesovec, Apr 08 2016 *)
A136405
Triangle read by rows: T(n,k) is the number of bi-partitions of the pair (n,k) into pairs (n_i,k_i) of positive integers such that sum k_i = k and sum n_i*k_i = n.
Original entry on oeis.org
1, 1, 2, 1, 1, 3, 1, 3, 2, 5, 1, 2, 4, 3, 7, 1, 4, 6, 7, 5, 11, 1, 3, 7, 8, 11, 7, 15, 1, 5, 8, 16, 14, 17, 11, 22, 1, 4, 12, 14, 23, 21, 25, 15, 30, 1, 6, 12, 24, 29, 38, 33, 37, 22, 42, 1, 5, 15, 24, 41, 42, 57, 47, 52, 30, 56, 1, 7, 18, 37, 47, 75, 68, 87, 70, 74, 42, 77
Offset: 1
Triangle begins:
1;
1, 2;
1, 1, 3;
1, 3, 2, 5;
1, 2, 4, 3, 7;
1, 4, 6, 7, 5, 11;
1, 3, 7, 8, 11, 7, 15;
1, 5, 8, 16, 14, 17, 11, 22;
1, 4, 12, 14, 23, 21, 25, 15, 30;
1, 6, 12, 24, 29, 38, 33, 37, 22, 42;
...
T(4,2) = 3 since (4,2) can be bi-partitioned as (2,2) or ((1,1),(3,1)) or ((2,1),(2,1)).
T(5,3) = 4 since (5,3) can be bi-partitioned as ((1,1),(2,2)) or ((3,1),(1,2)) or ((1,1),(1,1),(3,1)) or ((1,1),(2,1),(2,1)).
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P(k, w, n)={prod(i=1, k, 1 - x^(i*w) + O(x*x^(n-k*w)))}
T(n)={Vecrev(polcoef(prod(w=1, n, sum(k=0, n\w, (x*y)^(k*w)/P(k,w,n))), n)/y)}
{ for(n=1, 10, print(T(n))) } \\ Andrew Howroyd, Oct 23 2019
A267862
Number of planar lattice convex polygonal lines joining the origin and the point (n,n).
Original entry on oeis.org
1, 2, 5, 13, 32, 77, 178, 399, 877, 1882, 3959, 8179, 16636, 33333, 65894, 128633, 248169, 473585, 894573, 1673704, 3103334, 5705383, 10405080, 18831761, 33836627, 60378964, 107035022, 188553965, 330166814, 574815804, 995229598, 1714004131, 2936857097
Offset: 0
The two walks for n = 1 are
(0,0) -> (1,1)
(0,0) -> (1,0) -> (1,1).
The five possibilities for n = 2 are
(0,0) -> (2,2)
(0,0) -> (1,0) -> (2,1) -> (2,2)
(0,0) -> (1,0) -> (2,2)
(0,0) -> (2,0) -> (2,2)
(0,0) -> (2,1) -> (2,2).
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a[i_Integer, j_Integer, s_] := a[i, j, s] = If[i === 0, 1, Sum[a[i - x, j - y, y/x], {x, 1, i}, {y, Floor[s*x] + 1, j}]]; a[n_Integer] := a[n] = 1 + Sum[a[n - x, n - y, y/x], {x, 1, n}, {y, 0, x - 1}]; Flatten[{1, Table[a[n], {n, 30}]}]
nmax = 20; p = (1 - x)*(1 - y); Do[Do[p = Expand[p*If[GCD[i, j] == 1, (1 - x^i*y^j), 1]]; p = Select[p, (Exponent[#, x] <= nmax) && (Exponent[#, y] <= nmax) &], {i, 1, nmax}], {j, 1, nmax}]; p = Expand[Normal[Series[1/p, {x, 0, nmax}, {y, 0, nmax}]]]; p = Select[p, Exponent[#, x] == Exponent[#, y] &]; Flatten[{1, Table[Coefficient[p, x^n*y^n], {n, 1, nmax}]}] (* Vaclav Kotesovec, Apr 08 2016 *)
A302919
The number of ways of placing 2n-1 white balls and 2n-1 black balls into unlabeled bins such that each bin has both an odd number of white balls and black balls.
Original entry on oeis.org
1, 2, 4, 12, 32, 85, 217, 539, 1316, 3146, 7374, 16969, 38387, 85452, 187456, 405659, 866759, 1830086, 3821072, 7894447, 16148593, 32723147, 65719405, 130871128, 258513076, 506724988, 985968770, 1904992841, 3655873294, 6970687150, 13208622956, 24879427889, 46593011280, 86773920240, 160742462714, 296227087942, 543183754454, 991213989213
Offset: 1
For n = 3 the a(3) = 4 ways to place five white and five black balls are (wwwwwbbbbb), (wwwbbb)(wb)(wb), (wwwb)(wbbb)(wb), and (wb)(wb)(wb)(wb)(wb).
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nmax = 15; p = 1; Do[Do[p = Expand[p*(1 - x^(2*i - 1)*y^(2*j - 1))]; p = Select[p, (Exponent[#, x] <= 2*nmax - 1) && (Exponent[#, y] <= 2*nmax - 1) &], {i, 1, 2*nmax - 1}], {j, 1, nmax}]; p = Expand[Normal[Series[1/p, {x, 0, 2*nmax - 1}, {y, 0, 2*nmax - 1}]]]; p = Select[p, Exponent[#, x] == Exponent[#, y] &]; Flatten[{1, Table[Coefficient[p, x^(2*n - 1)*y^(2*n - 1)], {n, 1, nmax}]}] (* Vaclav Kotesovec, Apr 16 2018 *)
Showing 1-4 of 4 results.
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