A386932
Lexicographically earliest sequence of distinct positive integers that can be partitioned into runs of integers without common bits, the n-th such run having a(n) terms.
Original entry on oeis.org
1, 2, 4, 3, 8, 16, 32, 5, 10, 48, 6, 9, 64, 128, 256, 512, 1024, 2048, 7, 24, 96, 384, 1536, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 11, 20, 160, 320, 2560, 5120, 24576, 98304, 393216, 1572864, 6291456, 8388608
Offset: 1
The first terms and runs are:
n a(n) n-th run
- ---- -----------------------------------
1 1 1
2 2 2, 4
3 4 3, 8, 16, 32
4 3 5, 10, 48
5 8 6, 9, 64, 128, 256, 512, 1024, 2048
See
A385661 for a similar sequence.
A367177
Triangle read by rows, T(n, k) = [x^k] hypergeom([1/2, -n, -n], [1, 1], 4*x).
Original entry on oeis.org
1, 1, 2, 1, 8, 6, 1, 18, 54, 20, 1, 32, 216, 320, 70, 1, 50, 600, 2000, 1750, 252, 1, 72, 1350, 8000, 15750, 9072, 924, 1, 98, 2646, 24500, 85750, 111132, 45276, 3432, 1, 128, 4704, 62720, 343000, 790272, 724416, 219648, 12870
Offset: 0
Triangle T(n, k) starts:
[0] 1;
[1] 1, 2;
[2] 1, 8, 6;
[3] 1, 18, 54, 20;
[4] 1, 32, 216, 320, 70;
[5] 1, 50, 600, 2000, 1750, 252;
[6] 1, 72, 1350, 8000, 15750, 9072, 924;
[7] 1, 98, 2646, 24500, 85750, 111132, 45276, 3432;
[8] 1, 128, 4704, 62720, 343000, 790272, 724416, 219648, 12870;
[9] 1, 162, 7776, 141120, 1111320, 4000752, 6519744, 4447872, 1042470, 48620;
-
p := n -> hypergeom([1/2, -n, -n], [1, 1], 4*x):
T := (n, k) -> coeff(simplify(p(n)), x, k):
seq(seq(T(n, k), k = 0..n), n = 0..9);
A367025
Triangle read by rows, T(n, k) = [x^k] p(n), where p(n) = (1 - hypergeom([-1/2, -n - 1, -n - 1], [1, 1], 4*x)) / (2*x).
Original entry on oeis.org
1, 4, 1, 9, 9, 2, 16, 36, 32, 5, 25, 100, 200, 125, 14, 36, 225, 800, 1125, 504, 42, 49, 441, 2450, 6125, 6174, 2058, 132, 64, 784, 6272, 24500, 43904, 32928, 8448, 429, 81, 1296, 14112, 79380, 222264, 296352, 171072, 34749, 1430
Offset: 0
Triangle T(n, k) starts:
[0] 1;
[1] 4, 1;
[2] 9, 9, 2;
[3] 16, 36, 32, 5;
[4] 25, 100, 200, 125, 14;
[5] 36, 225, 800, 1125, 504, 42;
[6] 49, 441, 2450, 6125, 6174, 2058, 132;
[7] 64, 784, 6272, 24500, 43904, 32928, 8448, 429;
[8] 81, 1296, 14112, 79380, 222264, 296352, 171072, 34749, 1430;
[9] 100, 2025, 28800, 220500, 889056, 1852200, 1900800, 868725, 143000, 4862;
-
p := n -> (1 - hypergeom([-1/2, -n-1, -n-1], [1, 1], 4*x)) / (2*x):
T := (n, k) -> coeff(simplify(p(n)), x, k):
seq(seq(T(n, k), k = 0..n), n = 0..9);
-
T[n_,k_]:=Binomial[n+1,n-k]^2*Binomial[2*k,k]/(k+1);Flatten[Table[T[n,k],{n,0,9},{k,0,n}]] (* Detlef Meya, Nov 19 2023 *)
A367178
Triangle read by rows. T(n, k) = binomial(n, k)^2 * CatalanNumber(k).
Original entry on oeis.org
1, 1, 1, 1, 4, 2, 1, 9, 18, 5, 1, 16, 72, 80, 14, 1, 25, 200, 500, 350, 42, 1, 36, 450, 2000, 3150, 1512, 132, 1, 49, 882, 6125, 17150, 18522, 6468, 429, 1, 64, 1568, 15680, 68600, 131712, 103488, 27456, 1430, 1, 81, 2592, 35280, 222264, 666792, 931392, 555984, 115830, 4862
Offset: 0
Triangle T(n, k) starts:
[0] 1;
[1] 1, 1;
[2] 1, 4, 2;
[3] 1, 9, 18, 5;
[4] 1, 16, 72, 80, 14;
[5] 1, 25, 200, 500, 350, 42;
[6] 1, 36, 450, 2000, 3150, 1512, 132;
[7] 1, 49, 882, 6125, 17150, 18522, 6468, 429;
[8] 1, 64, 1568, 15680, 68600, 131712, 103488, 27456, 1430;
[9] 1, 81, 2592, 35280, 222264, 666792, 931392, 555984, 115830, 4862;
-
T := (n, k) -> binomial(n, k)^2 * binomial(2*k, k) / (k + 1):
seq(seq(T(n, k), k = 0..n), n = 0..9);
Showing 1-4 of 4 results.
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