|
|
A055232
|
|
Expansion of (1+2*x+3*x^2)/((1-x)^3*(1-x^2)).
|
|
9
|
|
|
1, 5, 16, 36, 69, 117, 184, 272, 385, 525, 696, 900, 1141, 1421, 1744, 2112, 2529, 2997, 3520, 4100, 4741, 5445, 6216, 7056, 7969, 8957, 10024, 11172, 12405, 13725, 15136, 16640, 18241, 19941, 21744, 23652, 25669, 27797, 30040, 32400, 34881, 37485, 40216, 43076
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
a(n) is the number of (w,x,y) having all terms in {0..n} and w <= floor((x+y)/2). - Clark Kimberling, Jun 02 2012
|
|
REFERENCES
|
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.28(c), y_3.
|
|
LINKS
|
|
|
FORMULA
|
G.f.: (1+2*x+3*x^2)/((1-x)^3*(1-x^2)).
a(n) = 3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4)-a(n-5). - Clark Kimberling, Jun 02 2012
a(n) = ((n+1)^3 + ceiling((n+1)/2)^2 + floor((n+1)/2)^2)/2. - Wesley Ivan Hurt, Apr 15 2016
E.g.f.: ((7 + 34*x + 26*x^2 + 4*x^3)*exp(x) + exp(-x))/8. - Ilya Gutkovskiy, Apr 16 2016
|
|
MAPLE
|
|
|
MATHEMATICA
|
Table[(7 + (-1)^n + 16*n + 14*n^2 + 4*n^3)/8, {n, 0, 50}] (* Wesley Ivan Hurt, Apr 15 2016 *)
LinearRecurrence[{3, -2, -2, 3, -1}, {1, 5, 16, 36, 69}, 40] (* Harvey P. Dale, Oct 25 2020 *)
|
|
PROG
|
(Magma) [(7+(-1)^n+16*n+14*n^2+4*n^3)/8 : n in [0..100]]; // Wesley Ivan Hurt, Apr 15 2016
(PARI) lista(nn) = for(n=0, nn, print1((7+(-1)^n+16*n+14*n^2+4*n^3)/8, ", ")); \\ Altug Alkan, Apr 16 2016
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|