A055232 Expansion of (1+2*x+3*x^2)/((1-x)^3*(1-x^2)).

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

Offset: 0

Author

N. J. A. Sloane, Jul 05 2000

Keywords

• nonn
• easy

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

First differences are in A212959. - Wesley Ivan Hurt, Apr 16 2016

References

• R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.28(c), y_3.

Links

• Michel Marcus, Table of n, a(n) for n = 0..999
• Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Crossrefs

Cf. A212959.

Programs

Magma

[(7+(-1)^n+16*n+14*n^2+4*n^3)/8 : n in [0..100]]; // Wesley Ivan Hurt, Apr 15 2016

Maple

A055232:=n->(7+(-1)^n+16*n+14*n^2+4*n^3)/8: seq(A055232(n), n=0..100); # Wesley Ivan Hurt, Apr 15 2016

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 *)

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

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) = (7+(-1)^n+16*n+14*n^2+4*n^3)/8. - Eric W. Weisstein, Apr 04 2013

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