A250304 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.

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, 63, 7, 784, 420, 84, 8, 1296, 612, 108, 9, 2025, 855, 135, 10, 3025, 1155, 165, 11, 4356, 1518, 198, 12, 6084, 1950, 234, 13, 8281, 2457, 273, 14, 11025, 3045, 315, 15, 14400, 3720, 360, 16

Offset: 1

Author

Derek Orr, Jan 15 2015

Keywords

• nonn
• easy
• tabf

Comments

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.

Examples

			Array starts:
n = 1:   0,   0,  0, 1;
n = 2:   1,   3,  3, 2;
n = 3:   9,  15,  9, 3;
n = 4:  36,  42, 18, 4;
n = 5: 100,  90, 30, 5;
n = 6: 225, 165, 45, 6;
n = 7: 441, 273, 63, 7;
n = 8: 784, 420, 84, 8;
...
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,5,0,0,0,-10,0,0,0,10,0,0,0,-5,0,0,0,1).

Crossrefs

Cf. A000537, A059270, A045943, A240970.

Programs

PARI

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

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

Formula

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

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

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

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

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

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