%I #49 Jan 03 2024 21:33:35

%S 1,5,16,43,106,249,568,1271,2806,6133,13300,28659,61426,131057,278512,

%T 589807,1245166,2621421,5505004,11534315,24117226,50331625,104857576,

%U 218103783,452984806,939524069,1946157028,4026531811,8321499106

%N Row sums of triangle A053218.

%C 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

%C 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

%C 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

%H Michael De Vlieger, <a href="/A053221/b053221.txt">Table of n, a(n) for n = 1..3311</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-13,12,-4).

%F a(n) = (n+2)*2^(n-1)-n-1. - _Vladeta Jovovic_, Feb 28 2003

%F G.f.: -x*(-1+x+x^2) / ( (2*x-1)^2*(x-1)^2 ). - _R. J. Mathar_, Sep 02 2011

%F a(n) = (1/2) * Sum_{k=1..n} Sum_{i=1..n} C(k,i) + C(n,k). - _Wesley Ivan Hurt_, Sep 22 2017

%F E.g.f.: exp(x)*(exp(x)-1)*(1+x). - _Enrique Navarrete_, Jul 25 2021

%F a(n+1) = 2*a(n) + A006127(n). - _Ya-Ping Lu_, Jan 01 2024

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

%p A053221 := proc(n) (n+2)*2^(n-1)-n-1 ; end proc: # _R. J. Mathar_, Sep 02 2011

%t Table[(n + 2)*2^(n - 1) - n - 1, {n, 29}] (* or *)

%t 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 *)

%t LinearRecurrence[{6,-13,12,-4},{1,5,16,43},30] (* _Harvey P. Dale_, Jun 28 2021 *)

%o (PARI) vector(50,n, (n+2)*2^(n-1)-n-1) \\ _G. C. Greubel_, Sep 03 2018

%o (Magma) [(n+2)*2^(n-1)-n-1: n in [1..50]]; // _G. C. Greubel_, Sep 03 2018

%Y Cf. A053218, A053219, A053220.

%Y Cf. A006127, A074909.

%K nonn,easy

%O 1,2

%A _Asher Auel_, Jan 01 2000