A238345
Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n where the k-th part is the first occurrence of a largest part, n>=1, 1<=k<=n.
Original entry on oeis.org
1, 2, 0, 3, 1, 0, 5, 2, 1, 0, 8, 5, 2, 1, 0, 14, 9, 6, 2, 1, 0, 24, 18, 12, 7, 2, 1, 0, 43, 33, 25, 16, 8, 2, 1, 0, 77, 62, 49, 35, 21, 9, 2, 1, 0, 140, 115, 95, 73, 49, 27, 10, 2, 1, 0, 256, 215, 181, 148, 108, 68, 34, 11, 2, 1, 0, 472, 401, 346, 291, 230, 158, 93, 42, 12, 2, 1, 0, 874, 753, 657, 569, 470, 353, 228, 125, 51, 13, 2, 1, 0
Offset: 1
Triangle starts:
01: 1;
02: 2, 0;
03: 3, 1, 0;
04: 5, 2, 1, 0;
05: 8, 5, 2, 1, 0;
06: 14, 9, 6, 2, 1, 0;
07: 24, 18, 12, 7, 2, 1, 0;
08: 43, 33, 25, 16, 8, 2, 1, 0;
09: 77, 62, 49, 35, 21, 9, 2, 1, 0;
10: 140, 115, 95, 73, 49, 27, 10, 2, 1, 0;
11: 256, 215, 181, 148, 108, 68, 34, 11, 2, 1, 0;
12: 472, 401, 346, 291, 230, 158, 93, 42, 12, 2, 1, 0;
13: 874, 753, 657, 569, 470, 353, 228, 125, 51, 13, 2, 1, 0;
14: 1628, 1416, 1250, 1102, 943, 753, 533, 324, 165, 61, 14, 2, 1, 0;
15: 3045, 2673, 2380, 2126, 1866, 1558, 1188, 791, 453, 214, 72, 15, 2, 1, 0;
...
-
g:= proc(n, m) option remember; `if`(n=0, 1,
add(g(n-j, min(n-j, m)), j=1..min(n, m)))
end:
h:= proc(n, t, m) option remember; `if`(n=0, 0,
`if`(t=1, add(g(n-j, j), j=m+1..n),
add(h(n-j, t-1, max(m, j)), j=1..n)))
end:
T:= (n, k)-> h(n, k, 0):
seq(seq(T(n, k), k=1..n), n=1..15);
-
g[n_, m_] := g[n, m] = If[n == 0, 1, Sum[g[n-j, Min[n-j, m]], {j, 1, Min[n, m]}]]; h[n_, t_, m_] := h[n, t, m] = If[n == 0, 0, If[t == 1, Sum[g[n-j, j], {j, m+1, n}], Sum[h[n-j, t-1, Max[m, j]], {j, 1, n}]]]; T[n_, k_] := h[n, k, 0]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 15}] // Flatten (* Jean-François Alcover, Jan 12 2015, translated from Maple *)
A238346
Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n where the k-th part is the last occurrence of a largest part, n>=1, 1<=k<=n.
Original entry on oeis.org
1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 5, 4, 1, 1, 8, 9, 8, 5, 1, 1, 14, 15, 15, 12, 6, 1, 1, 24, 27, 27, 24, 17, 7, 1, 1, 43, 47, 50, 46, 37, 23, 8, 1, 1, 77, 85, 90, 89, 75, 55, 30, 9, 1, 1, 140, 153, 165, 167, 152, 118, 79, 38, 10, 1, 1, 256, 279, 301, 313, 299, 250, 180, 110, 47, 11, 1, 1, 472, 511, 552, 582, 578, 516, 398, 267, 149, 57, 12, 1, 1
Offset: 1
Triangle starts:
01: 1,
02: 1, 1,
03: 2, 1, 1,
04: 3, 3, 1, 1,
05: 5, 5, 4, 1, 1,
06: 8, 9, 8, 5, 1, 1,
07: 14, 15, 15, 12, 6, 1, 1,
08: 24, 27, 27, 24, 17, 7, 1, 1,
09: 43, 47, 50, 46, 37, 23, 8, 1, 1,
10: 77, 85, 90, 89, 75, 55, 30, 9, 1, 1,
11: 140, 153, 165, 167, 152, 118, 79, 38, 10, 1, 1,
12: 256, 279, 301, 313, 299, 250, 180, 110, 47, 11, 1, 1,
13: 472, 511, 552, 582, 578, 516, 398, 267, 149, 57, 12, 1, 1,
...
A120643
Table T(n,k) = number of fractal initial sequences (where new values are successive integers) of length n whose last term is k.
Original entry on oeis.org
1, 1, 1, 2, 1, 1, 3, 2, 2, 1, 5, 4, 3, 3, 1, 8, 8, 5, 6, 4, 1, 14, 14, 10, 10, 10, 5, 1, 24, 25, 21, 16, 20, 15, 6, 1, 43, 43, 43, 28, 35, 35, 21, 7, 1, 77, 76, 83, 56, 57, 70, 56, 28, 8, 1, 140, 136, 153, 120, 93, 126, 126, 84, 36, 9, 1, 256, 248, 274, 256, 165, 211, 252, 210, 120, 45, 10, 1
Offset: 1
For n = 3, the 4 sequences are 1,1,1; 1,1,2; 1,2,1; and 1,2,3. Of these, 2 end in 1, 1 in 2 and 1 in 3, so row 3 is 2,1,1.
The table starts:
1
1,1
2,1,1
3,2,2,1
5,4,3,3,1
8,8,5,6,4,1
-
uppertrim[list_] := Fold[DeleteCases[#1, #2, 1, 1] &, list, Range[Max[list]]]; to[list_, 0] := Append[list, Part[list, Length[uppertrim@list] + 1]]; to[list_, 1] := Append[list, Max@list + 1]; allfractal[n_] := Fold[to[#1, #2] &, {1}, #] & /@ Tuples[{0, 1}, n]; k = 10; Flatten[Table[BinCounts[allfractal[k][[All, i]], {1, i + 1}] 2^(i - 1), {i, k + 1}]/2^k] (* Birkas Gyorgy, Nov 25 2012 *)
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