|
|
A201689
|
|
Number of involutions avoiding the pattern 21 (with a dot over the 1).
|
|
2
|
|
|
1, 0, 1, 1, 4, 9, 31, 94, 337, 1185, 4540, 17581, 71875, 299646, 1299637, 5760973, 26357764, 123241185, 591877543, 2902472734, 14571525145, 74613410169, 390197960716, 2078859419077, 11290463266843, 62400316038462, 351037047533581, 2007507147853429
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,5
|
|
COMMENTS
|
Baril gives a formula for a(n), but when I evaluate it I get A201687, not the values shown here.
|
|
LINKS
|
|
|
FORMULA
|
G.f.: 1/(G(0)+x), where G(k) = 1 - x - x^2*(k+1)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 15 2011
a(n) = A000085(n) - Sum_{r=0..n-1} a(r)*A000085(n-1-r). [from Baril, corrected]
|
|
EXAMPLE
|
G.f.: 1 + x^2 + x^3 + 4*x^4 + 9*x^5 + 31*x^6 + 94*x^7 + 337*x^8 + ...
|
|
MAPLE
|
b:= proc(n) option remember; `if`(n<1, 1, b(n-1)+(n-1)*b(n-2)) end:
a:= proc(n) option remember; b(n)-add(a(r)*b(n-1-r), r=0..n-1) end:
|
|
MATHEMATICA
|
b[n_] := b[n] = If[n < 1, 1, b[n - 1] + (n - 1)*b[n - 2]];
a[n_] := a[n] = b[n] - Sum[a[r]*b[n - 1 - r], {r, 0, n - 1}];
|
|
PROG
|
(PARI) seq(n)={my(g=serlaplace(exp(x+x^2/2 + O(x*x^n)))); Vec(g/(1 + x*g))} \\ Andrew Howroyd, Jan 10 2022
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|