Displaying 181-184 of 184 results found.
page
1
...
15
16
17
18
19
b(n, 2) where b(n, m) is defined by expansion of ((Product_{k>=1} (1 - x^(prime(n)*k))/(1 - x^k)^prime(n)) - 1)/prime(n) in powers of x.
+10
0
2, 3, 4, 5, 7, 8, 10, 11, 13, 16, 17, 20, 22, 23, 25, 28, 31, 32, 35, 37, 38, 41, 43, 46, 50, 52, 53, 55, 56, 58, 65, 67, 70, 71, 76, 77, 80, 83, 85, 88, 91, 92, 97, 98, 100, 101, 107, 113, 115, 116, 118, 121, 122, 127, 130, 133, 136, 137, 140, 142, 143, 148, 155, 157
COMMENTS
c(n, m) is defined by expansion of (Product_{k>=1} 1/(1 - x^k)^prime(n))/prime(n) in powers of x.
b(n, 2) = c(n, 2) for n > 1.
FORMULA
a(n) = A098090(n - 1) = (prime(n) + 3)/2 for n > 1.
Irregular triangle read by rows: T(n,k) = number of 1324-avoiding permutations of length n >= 0 having k >= 0 inversions.
+10
0
1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 5, 6, 5, 3, 1, 1, 2, 5, 10, 16, 20, 20, 15, 9, 4, 1, 1, 2, 5, 10, 20, 32, 51, 67, 79, 80, 68, 49, 29, 14, 5, 1, 1, 2, 5, 10, 20, 36, 61, 96, 148, 208, 268, 321, 351, 347, 308, 241, 165, 98, 49, 20, 6, 1, 1, 2, 5, 10, 20, 36, 65, 106, 171, 262, 397, 568, 784, 1019, 1264, 1478, 1628, 1681, 1619, 1441, 1173, 866, 574, 338, 174, 76, 27, 7, 1
REFERENCES
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 769, Fig. 12.13.
EXAMPLE
Triangle begins:
1;
1;
1, 1;
1, 2, 2, 1;
1, 2, 5, 6, 5, 3, 1;
1, 2, 5, 10, 16, 20, 20, 15, 9, 4, 1;
1, 2, 5, 10, 20, 32, 51, 67, 79, 80, 68, 49, 29, 14, 5, 1;
...
CROSSREFS
The rows appear to be converging to A000712.
Number of ordered pairs (a,g) with a in IS_n the symmetric inverse semigroup on [n] and g in symmetric group on [n] such that ag=ga.
+10
0
1, 2, 10, 60, 480, 4320, 46800, 554400, 7459200, 108864000, 1745452800, 30017433600, 558036864000, 11021826816000, 232330146048000, 5173159799808000, 121812482727936000, 3012672515973120000, 78301030421053440000, 2127572806150471680000, 60438151687124090880000
MAPLE
a:= proc(n) option remember; `if`(n=0, 1, 2*(n-1)!*
add(a(j)/j!*numtheory[sigma](n-j), j=0..n-1))
end:
MATHEMATICA
nn = 16; Table[Sum[PartitionsP[k] PartitionsP[n - k], {k, 0, n}], {n, 0, nn}] Table[n!, {n, 0, nn}]
Table read by antidiagonals: row n gives the Euler transform of the sequence (2,...,2,0,0,...) that contains n 2s followed by 0s.
+10
0
1, 1, 2, 1, 2, 3, 1, 2, 5, 4, 1, 2, 5, 8, 5, 1, 2, 5, 10, 14, 6, 1, 2, 5, 10, 18, 20, 7, 1, 2, 5, 10, 20, 30, 30, 8, 1, 2, 5, 10, 20, 34, 49, 40, 9, 1, 2, 5, 10, 20, 36, 59, 74, 55, 10, 1, 2, 5, 10, 20, 36, 63, 94, 110, 70, 11, 1, 2, 5, 10, 20, 36, 65, 104, 149, 158, 91, 12
EXAMPLE
Table begins:
| 0 1 2 3 4 5 6 7 8 9 10
--+----------------------------------
1 | 1 2 3 4 5 6 7 8 9 10 11
2 | 1 2 5 8 14 20 30 40 55 70 91
3 | 1 2 5 10 18 30 49 74 110 158 221
4 | 1 2 5 10 20 34 59 94 149 224 334
5 | 1 2 5 10 20 36 63 104 169 264 405
6 | 1 2 5 10 20 36 65 108 179 284 445
7 | 1 2 5 10 20 36 65 110 183 294 465
8 | 1 2 5 10 20 36 65 110 185 298 475
9 | 1 2 5 10 20 36 65 110 185 300 479
MATHEMATICA
Seed[i_, n_] := ConstantArray[2, i]~Join~ConstantArray[0, n - i];
A364842Table[n_] := Table[Seed[i, n] // EulerTransform, {i, 1, n}]
(*EulerTransform is defined in A005195*)
CROSSREFS
Analogous for initial 1s sequence A008284.
Search completed in 0.103 seconds
|