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A008705 Coefficient of x^n in (Product_{m=1..n}(1-x^m))^n. 8
1, -1, -1, 5, -5, -6, 11, 41, -125, -85, 1054, -2069, -209, 8605, -15625, 3990, 14035, 36685, -130525, -254525, 1899830, -3603805, -134905, 13479425, -25499225, 23579969, -64447293, 237487433, -133867445, -1795846200, 6309965146, -6788705842, -11762712973 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Degree of resulting polynomial is A002411(n). - Michel Marcus, Sep 05 2013

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..2856 (terms 0..256 from N. J. A. Sloane)

Morris Newman, Further identities and congruences for the coefficients of modular forms, Canadian J. Math 10 (1958): 577-586. See Table 1, column p=5.

Morris Newman, Further identities and congruences for the coefficients of modular forms [annotated scanned copy], Canadian J. Math 10 (1958): 577-586. See Table 1, column p=5.

FORMULA

a(n) = [x^n] exp(-n*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, May 30 2018

EXAMPLE

(1-x)^1 = -x + 1, hence a(1) = -1.

(1-x^2)^2*(1-x)^2 = x^6 - 2*x^5 - x^4 + 4*x^3 - x^2 - 2*x + 1, hence a(2) = -1.

MAPLE

C5:=proc(r) local t1, n; t1:=mul((1-x^n)^r, n=1..r+2); series(t1, x, r+1); coeff(%, x, r); end;

[seq(C5(i), i=0..30)]; # N. J. A. Sloane, Oct 04 2015

# second Maple program:

b:= proc(n, k) option remember; `if`(n=0, 1, -k*

      add(numtheory[sigma](j)*b(n-j, k), j=1..n)/n)

    end:

a:= n-> b(n$2):

seq(a(n), n=0..35);  # Alois P. Heinz, Jun 21 2018

MATHEMATICA

With[{m = 40}, Table[SeriesCoefficient[Series[(Product[1-x^j, {j, n}])^n, {x, 0, m}], n], {n, 0, m}]] (* G. C. Greubel, Sep 09 2019 *)

PROG

(PARI) a(n) = polcoeff(prod(m = 1, n, (1-x^m)^n), n); \\ Michel Marcus, Sep 05 2013

CROSSREFS

Bisections: A262308, A262309.

Main diagonal of A286354.

Sequence in context: A191695 A298915 A008945 * A327242 A173932 A249649

Adjacent sequences:  A008702 A008703 A008704 * A008706 A008707 A008708

KEYWORD

sign

AUTHOR

T. Forbes (anthony.d.forbes(AT)googlemail.com)

EXTENSIONS

More terms from Michel Marcus, Sep 05 2013

a(0)=1 prepended by N. J. A. Sloane, Oct 04 2015

STATUS

approved

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Last modified October 14 12:45 EDT 2019. Contains 328006 sequences. (Running on oeis4.)