A081892 Second binomial transform of C(n+2,2).

1, 5, 22, 90, 351, 1323, 4860, 17496, 61965, 216513, 747954, 2558790, 8680203, 29229255, 97785144, 325241892, 1076168025, 3544180029, 11622614670, 37967207922, 123587135991, 400980206115, 1297083797172, 4184141281200, 13462474572261, 43211719081593, 138390472875690

Offset

0,2

Comments

Binomial transform of A049611(n+1).

2nd binomial transform of C(n+2,2), A000217.

3rd binomial transform of (1,2,1,0,0,0,.....).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (9,-27,27).

Formula

a(n) = 3^(n - 2)*(n + 2)*(n + 9)/2 = 3^n*(n^2 + 11*n + 18)/18.

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

E.g.f.: (2 + 4*x + x^2)*exp(3*x)/2. - G. C. Greubel, Oct 18 2018

From Amiram Eldar, Feb 03 2026: (Start)

Sum_{n>=0} 1/a(n) = 20103579/980 - (354132/7)*log(3/2).

Sum_{n>=0} (-1)^n/a(n) = 14276709/980 - (354456/7)*log(4/3). (End)

Mathematica

LinearRecurrence[{9, -27, 27}, {1, 5, 22}, 50] (* G. C. Greubel, Oct 18 2018 *)

Prog

(PARI) x='x+O('x^30); Vec((1-2*x)^2/(1-3*x)^3) \\ G. C. Greubel, Oct 18 2018
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-2*x)^2/(1-3*x)^3)); // G. C. Greubel, Oct 18 2018

Crossrefs

Cf. A000217, A049611, A081893.

A right-edge column of triangle A024462.

Sequence in context: A086090 A037529 A108072 * A128566 A097138 A105467

Adjacent sequences: A081889 A081890 A081891 * A081893 A081894 A081895

Keyword

nonn,easy

Author

Paul Barry, Mar 30 2003

Status

approved