A333616 Expansion of x*(1 + 2*x + x^2 - 4*x^3 - x^4 + 2*x^5)/((1 - x)^3*(1 + x)^2).

0, 1, 3, 6, 6, 10, 10, 15, 15, 21, 21, 28, 28, 36, 36, 45, 45, 55, 55, 66, 66, 78, 78, 91, 91, 105, 105, 120, 120, 136, 136, 153, 153, 171, 171, 190, 190, 210, 210, 231, 231, 253, 253, 276, 276, 300, 300, 325, 325, 351, 351, 378, 378, 406, 406, 435, 435, 465, 465

Offset

0,3

Comments

For n > 0, a(n) is the n-th row sum of the triangle A333416.

Links

Table of n, a(n) for n=0..58.

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Formula

O.g.f.: x*(1 + 2*x + x^2 - 4*x^3 - x^4 + 2*x^5)/((1 - x)^3*(1 + x)^2).

E.g.f.: ((8 + 9*x + x^2)*cosh(x) + (15 + 7*x + x^2)*sinh(x) - 8*(1 + 2*x))/8.

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

a(n) = A008805(n+1) for n > 2.

Mathematica

LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 3, 6, 6, 10, 10, 15}, 59]

Prog

(PARI) my(x='x + O('x^59)); concat([0], Vec(serlaplace(((8 + 9*x + x^2)*cosh(x) + (15 + 7*x + x^2)*sinh(x) - 8*(1 + 2*x))/8)))
(Sage) (x*(1 + 2*x + x^2 - 4*x^3 - x^4 + 2*x^5)/((1 - x)^3*(1 + x)^2)).series(x, 59).coefficients(x, sparse=False)

Crossrefs

Cf. A000217, A008805, A333416.

Sequence in context: A159787 A184161 A276000 * A316563 A349212 A316140

Adjacent sequences: A333613 A333614 A333615 * A333617 A333618 A333619

Keyword

easy,nonn

Author

Stefano Spezia, Mar 29 2020

Status

approved