A250304 - OEIS

%I #18 Jun 13 2015 00:55:18

%S 0,0,0,1,1,3,3,2,9,15,9,3,36,42,18,4,100,90,30,5,225,165,45,6,441,273,

%T 63,7,784,420,84,8,1296,612,108,9,2025,855,135,10,3025,1155,165,11,

%U 4356,1518,198,12,6084,1950,234,13,8281,2457,273,14,11025,3045,315,15,14400,3720,360,16

%N Four-column array read by rows: T(n,k) = the coefficient of x^k in the expanded polynomial x^3 + (x+1)^3 + ... + (x+n-1)^3, for 0 <= k <= 3.

%C A240970 solves the Diophantine equation: k^3 + (k+1)^3 + ... + (k+n-1)^3 = y^3. This array gives the coefficients of the left hand side for specified n.

%H Colin Barker, <a href="/A250304/b250304.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_20">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,5,0,0,0,-10,0,0,0,10,0,0,0,-5,0,0,0,1).

%F a(4*k+1) = A000537(k), for k >= 0.

%F a(4*k+2) = A059270(k), for k >= 0.

%F a(4*k+3) = A045943(k), for k >= 0.

%F a(4*k) = k, for k >= 1.

%F a(n) = ((2*n^4+40*n^3+188*n^2-24*n-558-(2*n^4-24*n^3+188*n^2-792*n-558)*(-1)^n+(2*n^4-20*n^3-130*n^2+772*n+377)*(-1)^((2*n-1+(-1)^n)/4)-(2*n^4+40*n^3-196*n^2-280*n+594)*(-1)^((6*n-1+(-1)^n)/4)-(4*n^3+66*n^2-228*n-217)*(-1)^((10*n-1+(-1)^n)/4)))/8192. - _Luce ETIENNE_, May 22 2015

%F G.f.: x^4*(x^12-3*x^11+3*x^10-x^9-3*x^8+6*x^7-4*x^5+3*x^4-3*x^3-3*x^2-x-1) / ((x-1)^5*(x+1)^5*(x^2+1)^5). - _Colin Barker_, Jun 02 2015

%e Array starts:

%e n = 1:   0,   0,  0, 1;

%e n = 2:   1,   3,  3, 2;

%e n = 3:   9,  15,  9, 3;

%e n = 4:  36,  42, 18, 4;

%e n = 5: 100,  90, 30, 5;

%e n = 6: 225, 165, 45, 6;

%e n = 7: 441, 273, 63, 7;

%e n = 8: 784, 420, 84, 8;

%e ...

%o (PARI) for(n=1,50,for(k=0,3,print1(polcoeff(sum(i=1,n,(x+i-1)^3),k),", ")))

%o (PARI) concat([0,0,0], Vec(x^4*(x^12-3*x^11+3*x^10-x^9-3*x^8+6*x^7-4*x^5+3*x^4-3*x^3-3*x^2-x-1) / ((x-1)^5*(x+1)^5*(x^2+1)^5) + O(x^100))) \\ _Colin Barker_, Jun 02 2015

%Y Cf. A000537, A059270, A045943, A240970.

%K nonn,easy,tabf

%O 1,6

%A _Derek Orr_, Jan 15 2015