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

0,3

Comments

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

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).

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

Example

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 + ...

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 *)

Prog

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

Crossrefs

Cf. A133263.

Sequence in context: A002579 A023544 A133263 * A038088 A018917 A167385

Adjacent sequences: A238528 A238529 A238530 * A238532 A238533 A238534

Keyword

nonn,easy

Author

Michael Somos, Feb 28 2014

Status

approved