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A250303 Three-column array read by rows: T(n,k) = the coefficient of x^k in the expanded polynomial x^2 + (x+1)^2 + ... + (x+n-1)^2, for 0 <= k <= 2. 0

%I

%S 0,0,1,1,2,2,5,6,3,14,12,4,30,20,5,55,30,6,91,42,7,140,56,8,204,72,9,

%T 285,90,10,385,110,11,506,132,12,650,156,13,819,182,14,1015,210,15,

%U 1240,240,16,1496,272,17,1785,306,18,2109,342,19,2470,380,20,2870,420,21,3311,462,22,3795,506

%N Three-column array read by rows: T(n,k) = the coefficient of x^k in the expanded polynomial x^2 + (x+1)^2 + ... + (x+n-1)^2, for 0 <= k <= 2.

%C A001032 solves the Diophantine equation: k^2 + (k+1)^2 + ... + (k+n-1)^2 = y^2. This array gives the coefficients of the left hand side for specified n.

%F a(3*k+1) = A000330(k), for k >= 0.

%F a(3*k+2) = A002378(k), for k >= 0.

%F a(3*k) = k, for k >= 1.

%e Array starts:

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

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

%e n = 3: 5, 6, 3;

%e n = 4: 14, 12, 4;

%e n = 5: 30, 20, 5;

%e n = 6: 55, 30, 6;

%e n = 7: 91, 42, 7;

%e n = 8: 140, 56, 8;

%e ...

%o (PARI) for(n=1,50,for(k=0,2,print1(polcoeff(sum(i=1,n,(x+i-1)^2),k),", ")))

%Y Cf. A000330, A002378, A001032.

%K nonn,easy,tabf

%O 1,5

%A _Derek Orr_, Jan 15 2015

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Last modified October 17 13:28 EDT 2021. Contains 348049 sequences. (Running on oeis4.)