A053221 Row sums of triangle A053218.

1, 5, 16, 43, 106, 249, 568, 1271, 2806, 6133, 13300, 28659, 61426, 131057, 278512, 589807, 1245166, 2621421, 5505004, 11534315, 24117226, 50331625, 104857576, 218103783, 452984806, 939524069, 1946157028, 4026531811, 8321499106

Offset: 1

Author

Asher Auel, Jan 01 2000

Keywords

• nonn
• easy

Comments

Considered as a vector, the sequence = A074909 * [1, 2, 3, ...], where A074909 is the beheaded Pascal's triangle as a matrix. - Gary W. Adamson, Mar 06 2012

a(n) is the sum of the upper left n X n subarray of A052509 (viewed as an infinite square array). For example (1+1+1) + (1+2+2) + (1+3+4) = 16. - J. M. Bergot, Nov 06 2012

Number of ternary strings of length n that contain at least one 2 and at most one 0. For example, a(3) = 16 since the strings are the 6 permutations of 201, the 3 permutations of 211, the 3 permutations of 220, the 3 permutations of 221, and 222. - Enrique Navarrete, Jul 25 2021

Examples

			a(4) = 4 + 7 + 12 + 20 = 43.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..3311
• Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Crossrefs

Cf. A053218, A053219, A053220.

Cf. A006127, A074909.

Programs

Magma

[(n+2)*2^(n-1)-n-1: n in [1..50]]; // G. C. Greubel, Sep 03 2018

Maple

A053221 := proc(n) (n+2)*2^(n-1)-n-1 ; end proc: # R. J. Mathar, Sep 02 2011

Mathematica

Table[(n + 2)*2^(n - 1) - n - 1, {n, 29}] (* or *)
Rest@ CoefficientList[Series[-x (-1 + x + x^2)/((2 x - 1)^2*(x - 1)^2), {x, 0, 29}], x] (* Michael De Vlieger, Sep 22 2017 *)
LinearRecurrence[{6,-13,12,-4},{1,5,16,43},30] (* Harvey P. Dale, Jun 28 2021 *)

PARI

vector(50,n, (n+2)*2^(n-1)-n-1) \\ G. C. Greubel, Sep 03 2018

Formula

a(n) = (n+2)*2^(n-1)-n-1. - Vladeta Jovovic, Feb 28 2003

G.f.: -x*(-1+x+x^2) / ( (2*x-1)^2*(x-1)^2 ). - R. J. Mathar, Sep 02 2011

a(n) = (1/2) * Sum_{k=1..n} Sum_{i=1..n} C(k,i) + C(n,k). - Wesley Ivan Hurt, Sep 22 2017

E.g.f.: exp(x)*(exp(x)-1)*(1+x). - Enrique Navarrete, Jul 25 2021

a(n+1) = 2*a(n) + A006127(n). - Ya-Ping Lu, Jan 01 2024