A188671 A000145(n) / 8 - (n^5 + 1).

1, 0, -24, -32, 108, 275, -176, -1056, 45, 3157, 1080, -6541, -836, 16839, 2072, -33824, 1188, 67100, 1672, -95883, 19162, 161083, -8208, -224653, 2707, 371325, 67500, -520025, -1188, 870551, 8512, -1082400, 148334, 1419889, 10428, -1588228

Offset: 1

Author

Michael Somos, Apr 11 2011

Keywords

• sign

Comments

Theorem 2 in the Hales reference defines t_p = (n_p - 8(p^5 + 1)) / (32 p^(5/2)) where n_p is the number of ways to express p as a sum of 12 squares.

Examples

			x - 24*x^3 - 32*x^4 + 108*x^5 + 275*x^6 - 176*x^7 - 1056*x^8 + 45*x^9 + ...
		

Links

• T. C. Hales, The Mathematical Work of the 2010 Fields Medalists, Notices Amer. Math. Soc, 58 (No. 3, Mar 2011), 453-457. See p. 457, Theorem 2.

Crossrefs

Cf. A000145.

Programs

PARI

{a(n) = if( n<1, 0, polcoeff( sum( k = 1, sqrtint(n), 2 * x^k^2, 1 + x*O(x^n))^12, n) / 8 - (n^5 + 1))}

Formula

G.f.: ((Sum_{k} x^k^2)^12 - 1) / 8 - (2*x + 21*x^2 + 76*x^3 + 16*x^4 + 6*x^5 - x^6) / (1 - x)^6.

a(n) = A000145(n) / 8 - (n^5 + 1).