A120815
Number of permutations of length n with exactly 7 occurrences of the pattern 2-13.
Original entry on oeis.org
42, 1664, 33338, 468200, 5253864, 50442128, 431645370, 3380738400, 24682378500, 170201240352, 1119398566704, 7074531999584, 43215257135312, 256343213520000, 1482127305153560, 8378542979807616, 46428426576857886
Offset: 7
Robert Parviainen (robertp(AT)ms.unimelb.edu.au), Jul 06 2006; definition corrected Feb 08 2008
- R. Parviainen, Lattice path enumeration of permutations with k occurrences of the pattern 2-13, preprint, 2006.
- Robert Parviainen, Lattice Path Enumeration of Permutations with k Occurrences of the Pattern 2-13, Journal of Integer Sequences, Vol. 9 (2006), Article 06.3.2.
A264496
Number of permutations of [n] with exactly nine (possibly overlapping) occurrences of the generalized pattern 13-2.
Original entry on oeis.org
1, 256, 11115, 258564, 4235959, 55145400, 609428105, 5954337792, 52838734860, 433992926048, 3345094317796, 24447556164496, 170782982829160, 1147577406468288, 7455096381483206, 47017233051466444, 288854847620573814, 1733658007070671968, 10189610103251200265
Offset: 7
-
b:= proc(u, o) option remember; `if`(u+o=0, 1, add(
b(u-j, o+j-1), j=1..u) +add(convert(series(
b(u+j-1, o-j)*x^(j-1), x, 10),polynom), j=1..o))
end:
a:= n-> coeff(b(n, 0), x, 9):
seq(a(n), n=7..25);
A264497
Number of permutations of [n] with exactly ten (possibly overlapping) occurrences of the generalized pattern 13-2.
Original entry on oeis.org
74, 5343, 166144, 3342042, 51052432, 643569953, 7034981562, 68865924682, 617271459112, 5148104341380, 40431758240544, 301784671671496, 2156289904582398, 14834095711273034, 98718321568669660, 637965526671540768, 4016617646502383422, 24704067590177063445
Offset: 8
-
b:= proc(u, o) option remember; `if`(u+o=0, 1, add(
b(u-j, o+j-1), j=1..u) +add(convert(series(
b(u+j-1, o-j)*x^(j-1), x, 11),polynom), j=1..o))
end:
a:= n-> coeff(b(n, 0), x, 10):
seq(a(n), n=8..25);
A287328
Number of permutations of [n] with exactly n (possibly overlapping) occurrences of the generalized pattern 13-2 (alternatively: 2-13, 2-31, or 31-2).
Original entry on oeis.org
1, 0, 0, 0, 0, 0, 2, 42, 716, 11115, 166144, 2440438, 35573362, 517145226, 7517489564, 109427233994, 1596263948916, 23344423478564, 342325226476816, 5033781463313268, 74223067319020872, 1097343555359525542, 16265399526715750734, 241690082996256941332
Offset: 0
-
b:= proc(u, o) option remember;
`if`(u+o=0, 1, add(b(u-j, o+j-1), j=1..u)+
add(expand(b(u+j-1, o-j)*x^(j-1)), j=1..o))
end:
a:= n-> coeff(b(n, 0), x, n):
seq(a(n), n=0..30);
-
b[u_, o_] := b[u, o] = If[u + o == 0, 1, Sum[b[u - j, o + j - 1], {j, 1, u}] + Sum[Expand[b[u + j - 1, o - j]*x^(j - 1)], {j, 1, o}]];
a[n_] := Coefficient[b[n, 0], x, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 06 2023, after Alois P. Heinz *)
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