A010818 Expansion of Product (1 - x^k)^10 in powers of x.

1, -10, 35, -30, -105, 238, 0, -260, -165, 140, 1054, -770, -595, 0, -715, 2162, 455, 0, -2380, -1820, 2401, -680, 1495, 3080, 1615, -6958, -1925, 0, 0, 5100, -1442, 8330, -5355, 1330, 0, -16790, 0, 8190, 8265, 0, 1918, 0, 8415, -10230, -7140, -9362

Offset

0,2

References

Newman, Morris; A table of the coefficients of the powers of eta(tau). Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389.

S. Cooper, The Quintuple product identity, Int. J. Number Theory 2 (2006), no. 1, 115-161. See after equation (73).

S. Cooper, M. D. Hirschhorn and R. Lewis, Powers of Euler's Product and Related Identities, The Ramanujan Journal, Vol. 4 (2), 137-155 (2000).

S. R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.

Formula

Expansion of f(-x)^10 in powers of x where f() is a Ramanujan theta function.

Expansion of q^(-5/12) * eta(q)^10 in powers of q. - Michael Somos, Jun 09 2011

a(n) = b(12*n + 5) / 48 where b() is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * p^(2*e) if p == 7 or 11 (mod 12), b(p^e) = b(p) * b(p^(e-1)) - p^4 * b(p^(e-2)) if p == 1 or 5 (mod 12). - Michael Somos, Jun 24 2013

G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 12^5 (t/i)^5 f(t) where q = exp(2 Pi i t). - Michael Somos, Jan 06 2014

G.f.: Product_{k>0} (1 - x^k)^10. a(49*n + 20) = 2401 * a(n).

48 * a(n) = A234565(3*n + 1). a(7*n + 2) = 0 unless n == 2 (mod 7). - Michael Somos, Jul 18 2014

a(0) = 1, a(n) = -(10/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 27 2017

G.f.: exp(-10*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 05 2018

Let M = p_1*...*p_k be a positive integer whose prime factors p_i (not necessarily distinct) are congruent to 7 (mod 12) or 11 (mod 12). Then a( M^2*n + 10*(M^2 - 1)/24 ) = M^4*a(n). See Cooper et al., Theorem 1. - Peter Bala, Dec 01 2020

Example

G.f. = 1 - 10*x + 35*x^2 - 30*x^3 - 105*x^4 + 238*x^5 - 260*x^7 - 165*x^8 + ...
G.f. = q^5 - 10*q^17 + 35*q^29 - 30*q^41 - 105*q^53 + 238*q^65 - 260*q^89 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^10, {x, 0, n}]; (* Michael Somos, Jun 24 2013 *)

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( eta(x + x * O(x^n))^10, n))}; /* Michael Somos, Jun 09 2011 */
(PARI) {a(n) = local(m, x, y, z); if( n<0, 0, m = 12*n + 5; z = 0; for( x = -sqrtint(m), sqrtint(m), if( x%6 != 1, next); if( issquare( m - x^2, &y), if( y%6 == 2, y = -y); if( y%6 == 4, z += x*y * (x*x - y*y) ))); z / 6)}; /* Michael Somos, Jun 09 2011 */
(PARI) {a(n) = local(A, p, e, i, x, y, a0, a1); if( n<0, 0, n = 12*n + 5; A = factor(n); 1 / 48 * prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p<5, 0, if( p%12 > 6, if( e%2, 0, p^(2*e)), forstep( i = 1, sqrtint( p), 2, if( issquare( p - i^2, &y), x=i; break)); if( p%12 == 5, a1 = 8 * x*y * (x-y) * (x+y) * (-1)^((x%6==1) + (y%6==4)), a1 = 2 * (x^2-y^2+2*x*y) * (x^2-y^2-2*x*y) * (-1)^(x%6==3) ); a0 = 1; y = a1; for( i=2, e, x = y * a1 - p^4 * a0; a0=a1; a1=x); a1 )))))}; /* Michael Somos, Jun 24 2013 */

Crossrefs

Cf. A122266, A234565.

Powers of Euler's product: A000594, A000727 - A000731, A000735, A000739, A002107, A010815 - A010840.

Sequence in context: A230895 A254674 A260336 * A243939 A065195 A115149

Adjacent sequences: A010815 A010816 A010817 * A010819 A010820 A010821

Keyword

sign,easy

Author

N. J. A. Sloane

Status

approved