A120909
Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).
Original entry on oeis.org
3, 3, 6, 3, 12, 12, 3, 18, 36, 24, 3, 24, 72, 96, 48, 3, 30, 120, 240, 240, 96, 3, 36, 180, 480, 720, 576, 192, 3, 42, 252, 840, 1680, 2016, 1344, 384, 3, 48, 336, 1344, 3360, 5376, 5376, 3072, 768, 3, 54, 432, 2016, 6048, 12096, 16128, 13824, 6912, 1536, 3, 60
Offset: 1
T(3,2)=12 because we have 001,002,011,022,100,110,112,122,200,211,220 and 221.
Triangle starts:
3;
3, 6;
3, 12, 12;
3, 18, 36, 24;
3, 24, 72, 96, 48;
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T:=(n,k)->3*2^(k-1)*binomial(n-1,k-1): for n from 1 to 11 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
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nn=15;f[list_]:=Select[list,#>0&];a=y x/(1-x) +1;b=a^2/(1-(a-1)^2 );Drop[Map[f,CoefficientList[Series[b a/(1-(a-1)(b-1)),{x,0,nn}],{x,y}]],1]//Grid (* Geoffrey Critzer, Nov 20 2012 *)
A380747
Array read by ascending antidiagonals: A(n,k) = [x^n] (1 - x)/(1 - k*x)^2.
Original entry on oeis.org
1, -1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 8, 5, 1, 0, 1, 20, 21, 7, 1, 0, 1, 48, 81, 40, 9, 1, 0, 1, 112, 297, 208, 65, 11, 1, 0, 1, 256, 1053, 1024, 425, 96, 13, 1, 0, 1, 576, 3645, 4864, 2625, 756, 133, 15, 1, 0, 1, 1280, 12393, 22528, 15625, 5616, 1225, 176, 17, 1
Offset: 0
The array begins as:
1, 1, 1, 1, 1, 1, ...
-1, 1, 3, 5, 7, 9, ...
0, 1, 8, 21, 40, 65, ...
0, 1, 20, 81, 208, 425, ...
0, 1, 48, 297, 1024, 2625, ...
0, 1, 112, 1053, 4864, 15625, ...
0, 1, 256, 3645, 22528, 90625, ...
...
Cf.
A000012 (k=1 or n=0),
A000567 (n=2),
A001792 (k=2),
A007778,
A060747 (n=1),
A081038 (k=3),
A081039 (k=4),
A081040 (k=5),
A081041 (k=6),
A081042 (k=7),
A081043 (k=8),
A081044 (k=9),
A081045 (k=10),
A103532,
A154955,
A380748 (antidiagonal sums).
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A[0,0]:=1; A[1,0]:=-1; A[n_,k_]:=((k-1)*n+k)k^(n-1); Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten (* or *)
A[n_,k_]:=SeriesCoefficient[(1-x)/(1-k*x)^2,{x,0,n}]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten (* or *)
A[n_,k_]:=n!SeriesCoefficient[Exp[k*x](1+(k-1)*x),{x,0,n}]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten
A183189
Triangle T(n,k), read by rows, given by (2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Original entry on oeis.org
1, 2, 0, 6, 1, 0, 18, 5, 0, 0, 54, 21, 1, 0, 0, 162, 81, 8, 0, 0, 0, 486, 297, 45, 1, 0, 0, 0, 1458, 1053, 216, 11, 0, 0, 0, 0, 4374, 3645, 945, 78, 1, 0, 0, 0, 0, 13122, 12393, 3888, 450, 14, 0, 0, 0, 0, 0
Offset: 0
Triangle begins:
1
2, 0
6, 1, 0
18, 5, 0, 0
54, 21, 1, 0, 0
162, 81, 8, 0, 0, 0
486, 297, 45, 1, 0, 0, 0
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