A158501 Hankel transform of A158500.

1, 0, 25, -24, 105, -104, 273, -272, 561, -560, 1001, -1000, 1625, -1624, 2465, -2464, 3553, -3552, 4921, -4920, 6601, -6600, 8625, -8624, 11025, -11024, 13833, -13832, 17081, -17080, 20801, -20800, 25025, -25024, 29785, -29784, 35113, -35112, 41041, -41040

Offset

0,3

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (-1,3,3,-3,-3,1,1).

Formula

G.f.: (1+x+22*x^2-2*x^3+9*x^4+x^5) / ((1-x)^3*(1+x)^4).

a(n) = -a(n-1) +3*a(n-2) +3*a(n-3) -3*a(n-4) -3*a(n-5) +a(n-6) +a(n-7).

From Colin Barker, Jan 29 2016: (Start)

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

a(n) = (2*n^3 + 9*n^2 + 10*n + 3)/3 for n even.

a(n) = (-2*n^3 - 3*n^2 + 2*n + 3)/3 for n odd. (End)

From G. C. Greubel, Mar 17 2025: (Start)

a(n) = 4*(-1)^n*A000292(n) + A000290(n+1).

E.g.f.: (1/3)*( 3*(1+3*x+x^2)*exp(x) - 2*x*(6-6*x+x^2)*exp(-x) ). (End)

Mathematica

LinearRecurrence[{-1, 3, 3, -3, -3, 1, 1}, {1, 0, 25, -24, 105, -104, 273}, 40] (* Harvey P. Dale, Aug 19 2012 *)

Prog

(PARI) Vec((1+x+22*x^2-2*x^3+9*x^4+x^5)/((1-x)^3*(1+x)^4) + O(x^50)) \\ Colin Barker, Jan 29 2016
(Magma)
A158501:= func< n | 4*(-1)^n*Binomial(n+2, 3) + (n+1)^2 >;
[A158501(n): n in [0..40]]; // G. C. Greubel, Mar 17 2025
(SageMath)
def A158501(n): return 4*(-1)^n*binomial(n+2, 3) + (n+1)^2
print([A158501(n) for n in range(41)]) # G. C. Greubel, Mar 17 2025

Crossrefs

Cf. A000290, A000292, A158500.

Sequence in context: A061438 A022981 A023467 * A330272 A194219 A291434

Adjacent sequences: A158498 A158499 A158500 * A158502 A158503 A158504

Keyword

easy,sign

Author

Paul Barry, Mar 20 2009

Status

approved