A238531 Expansion of (1 - x + x^2)^2 / (1 - x)^3 in powers of x.

1, 1, 3, 5, 8, 12, 17, 23, 30, 38, 47, 57, 68, 80, 93, 107, 122, 138, 155, 173, 192, 212, 233, 255, 278, 302, 327, 353, 380, 408, 437, 467, 498, 530, 563, 597, 632, 668, 705, 743, 782, 822, 863, 905, 948, 992, 1037, 1083, 1130, 1178, 1227, 1277, 1328, 1380

Offset: 0

Author

Michael Somos, Feb 28 2014

Keywords

• nonn
• easy

Comments

Essentially the same as A152948, A133263 and A089071. - R. J. Mathar, Mar 30 2014

Examples

			G.f. = 1 + x + 3*x^2 + 5*x^3 + 8*x^4 + 12*x^5 + 17*x^6 + 23*x^7 + 30*x^8 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A133263.

Programs

Magma

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x+x^2)^2/(1-x)^3)); // G. C. Greubel, Aug 07 2018

Mathematica

a[ n_] := (n^2 - n) / 2 + If[ n == 0 || n == 1, 1, 2];
CoefficientList[Series[(1-x+x^2)^2/(1-x)^3, {x, 0, 50}], x] (* G. C. Greubel, Aug 07 2018 *)

PARI

{a(n) = (n^2 - n) / 2 + 2 - (n==0) - (n==1)};

PARI

{a(n) = if( n<0, n = 1-n); polcoeff( (1 - x + x^2)^2 / (1 - x)^3 + x * O(x^n), n)};

Formula

Euler transform of length 6 sequence [1, 2, 2, 0, 0, -2].

Binomial transform of [1, 0, 2, -2, 3, -4, 5, -6, ...].

a(n) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n.

G.f.: (1 - x + x^2)^2 / (1 - x)^3.

a(n) = a(1 - n) for all n in Z.

a(n + 1) = A133263(n) if n>=0. a(n) = (n^2 - n) / 2 + 2 unless n=0 or n=1.

(1 + x^2 + x^3 + x^4 + ...)*(1 + x + 2x^2 + 3x^3 + 4x^4 + ...) = (1 + x + 3x^2 + 5x^3 + 8x^4 + 12x^5 + ...). - Gary W. Adamson, Jul 27 2010