A208657
Triangular array read by rows: n*binomial(n,n-k+1)-binomial(n-1,n-k) with k = 1..n, n >= 1.
Original entry on oeis.org
0, 1, 3, 2, 7, 8, 3, 13, 21, 15, 4, 21, 44, 46, 24, 5, 31, 80, 110, 85, 35, 6, 43, 132, 225, 230, 141, 48, 7, 57, 203, 413, 525, 427, 217, 63, 8, 73, 296, 700, 1064, 1078, 728, 316, 80, 9, 91, 414, 1116, 1974, 2394, 2016, 1164, 441, 99, 10, 111, 560, 1695
Offset: 1
Triangle begins:
0,
1, 3,
2, 7, 8,
3, 13, 21, 15,
4, 21, 44, 46, 24,
5, 31, 80, 110, 85, 35,
6, 43, 132, 225, 230, 141, 48,
7, 57, 203, 413, 525, 427, 217, 63,
8, 73, 296, 700, 1064, 1078, 728, 316, 80,
9, 91, 414, 1116, 1974, 2394, 2016, 1164, 441, 99;
...
-
[n*Binomial(n,n-k+1)-Binomial(n-1,n-k): k in [1..n], n in [1..11]]; // Bruno Berselli, Apr 15 2015
-
z = 12;
f[n_, k_] := n*Binomial[n, k] - Binomial[n - 1, k - 1]
t = Table[f[n, k], {n, 1, z}, {k, 1, n}];
TableForm[t] (* A208656 as a triangle *)
Flatten[t] (* A208656 as a sequence *)
r = Table[f[n, k], {n, 1, z}, {k, n, 1, -1}];
TableForm[r] (* A208657 as a triangle *)
Flatten[r] (* A208657 as a sequence *)
Table[Sum[f[n, k], {k, 1, n}], {n, 1, 3 z}](* A208658 *)
A208656
Triangle T(n, k) = n*C(n,k) - C(n-1,k-1), 1 <= k <= n, read by rows.
Original entry on oeis.org
0, 3, 1, 8, 7, 2, 15, 21, 13, 3, 24, 46, 44, 21, 4, 35, 85, 110, 80, 31, 5, 48, 141, 230, 225, 132, 43, 6, 63, 217, 427, 525, 413, 203, 57, 7, 80, 316, 728, 1078, 1064, 700, 296, 73, 8, 99, 441, 1164, 2016, 2394, 1974, 1116, 414, 91, 9, 120, 595, 1770
Offset: 1
First five rows:
0
3....1
8....7....2
15...21...13...3
24...46...44...21...4
-
z = 12;
f[n_, k_] := n*Binomial[n, k] - Binomial[n - 1, k - 1]
t = Table[f[n, k], {n, 1, z}, {k, 1, n}];
TableForm[t] (* A208656 as a triangle *)
Flatten[t] (* A208656 as a sequence *)
r = Table[f[n, k], {n, 1, z}, {k, n, 1, -1}];
TableForm[r] (* A208657 as a triangle *)
Flatten[r] (* A208657 as a sequence *)
Table[Sum[f[n, k], {k, 1, n}], {n, 1, 3 z}](* A208658 *)
A367468
Triangle read by rows: T(n,k) is the total number of movable letters in all members of the k-partitions of [n], with 1 <= k <= n.
Original entry on oeis.org
0, 1, 0, 2, 4, 0, 3, 17, 9, 0, 4, 52, 68, 16, 0, 5, 139, 345, 190, 25, 0, 6, 346, 1474, 1440, 430, 36, 0, 7, 825, 5733, 8904, 4550, 847, 49, 0, 8, 1912, 21048, 49056, 38304, 11928, 1512, 64, 0, 9, 4343, 74385, 251250, 282135, 130998, 27342, 2508, 81, 0
Offset: 1
Triangle begins:
0;
1, 0;
2, 4, 0;
3, 17, 9, 0;
4, 52, 68, 16, 0;
5, 139, 345, 190, 25, 0;
...
-
T[n_,k_]:=If[k==1,n-1,(2n-1)StirlingS2[n,k]/2-StirlingS2[n+1,k]/2+StirlingS2[n-1,k-2]/2]; Table[T[n,k],{n,10},{k,n}]//Flatten
Showing 1-3 of 3 results.
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