cp's OEIS Frontend

A259181 a(n) = n*(n+1)*(n+2)*(n+3)*(2*n^2+6*n+7)/360.

%I A259181 #64 Jan 28 2025 10:52:18
%S A259181 0,1,9,43,147,406,966,2058,4026,7359,12727,21021,33397,51324,76636,
%T A259181 111588,158916,221901,304437,411103,547239,719026,933570,1198990,
%U A259181 1524510,1920555,2398851,2972529,3656233,4466232,5420536,6539016,7843528,9358041,11108769
%N A259181 a(n) = n*(n+1)*(n+2)*(n+3)*(2*n^2+6*n+7)/360.
%C A259181 After 0, second bisection of A129548.
%C A259181 This sequence is also the total number of squares of all sizes in i X i subsquares in an n X n grid, whereas A000330 simply gives the number of all sizes of squares in an n X n grid. See illustrations.
%H A259181 Colin Barker, <a href="/A259181/b259181.txt">Table of n, a(n) for n = 0..1000</a>
%H A259181 Luce ETIENNE, <a href="/A259181/a259181.pdf">illustration of initial terms</a>
%H A259181 Feihu Liu, Guoce Xin, and Chen Zhang, <a href="https://arxiv.org/abs/2412.18744">Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS</a>, arXiv:2412.18744 [math.CO], 2024. See p. 14.
%H A259181 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A259181 a(n) = (1/6)*Sum_{i=0..n} (i+1)*(i+2)*(2*i+3)*(n-i)^2.
%F A259181 a(n) = Sum_{i=0..n} A000290(n-i)*A000330(i+1).
%F A259181 G.f.: x*(1 + x)^2 / (1 - x)^7. - _Colin Barker_, Nov 08 2015
%F A259181 a(n) = (A000539(n+1) - A000217(n+1))/30. - _Yasser Arath Chavez Reyes_, Feb 24 2024
%e A259181 a(0) = 0; a(1) = 1*1; a(2) = 4*1+1*5 = 9; a(3) = 9*1+4*5+1*14 = 43.
%o A259181 (PARI) vector(100, n, n--; n*(n+1)*(n+2)*(n+3)*(2*n^2+6*n+7)/360) \\ _Altug Alkan_, Nov 08 2015
%o A259181 (PARI) concat(0, Vec(-x*(x+1)^2 / (x-1)^7 + O(x^100))) \\ _Colin Barker_, Nov 08 2015
%Y A259181 Cf. A000217, A000290, A000330, A000539, A129548.
%Y A259181 Cf. A060060: (1/6)*Sum_{i=0..n} (i+1)*(i+2)*(2*i+3)*i^2.
%K A259181 nonn,easy
%O A259181 0,3
%A A259181 _Luce ETIENNE_, Nov 08 2015