A178947 Expansion of x*(1+2*x+8*x^2+4*x^3+3*x^4) / ( (1+x)^2*(x-1)^4 ).

1, 4, 17, 38, 81, 138, 229, 340, 497, 680, 921, 1194, 1537, 1918, 2381, 2888, 3489, 4140, 4897, 5710, 6641, 7634, 8757, 9948, 11281, 12688, 14249, 15890, 17697, 19590, 21661, 23824, 26177, 28628, 31281, 34038, 37009, 40090, 43397, 46820, 50481, 54264, 58297

Offset: 1

Author

Gary W. Adamson, Dec 30 2010

Keywords

• nonn
• easy

Comments

Let S(x) be the generating function of A016777; then the generating function of this sequence is x/2 * (S(x)^2 + S(x^2)): the sequence is obtained by adding half of the convolution square, A100175, and the aerated A016777.

Examples

			(1/2) * ((1, 8, 30, 76, 155, 276,...) + (1, 0, 4, 0, 7, 0, 10,...)) = (1, 4, 17, 38, 81, 138, 229,...).
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Crossrefs

Cf. A100175, A016777.

Programs

Mathematica

LinearRecurrence[{2,1,-4,1,2,-1},{1,4,17,38,81,138},50] (* Harvey P. Dale, Jun 12 2018 *)

Formula

a(2n) = A100175(2n)/2.

a(2n+1) = (A100175(2n+1)+A016777(n))/2.

From Colin Barker, Aug 02 2016: (Start)

a(n) = (-1+(-1)^n+(7-3*(-1)^n)*n-6*n^2+6*n^3)/8.

a(n) = (3*n^3-3*n^2+2*n)/4 for n even.

a(n) = (3*n^3-3*n^2+5*n-1)/4 for n odd.

a(n) = 2*a(n-1)+a(n-2)-4*a(n-3)+a(n-4)+2*a(n-5)-a(n-6) for n>6.

(End)