|
%I #61 Apr 20 2023 11:51:06
%S 1,-1,-1,5,-5,-6,11,41,-125,-85,1054,-2069,-209,8605,-15625,3990,
%T 14035,36685,-130525,-254525,1899830,-3603805,-134905,13479425,
%U -25499225,23579969,-64447293,237487433,-133867445,-1795846200,6309965146,-6788705842,-11762712973
%N Coefficient of x^n in (Product_{m=1..n}(1-x^m))^n.
%C Degree of resulting polynomial is A002411(n). - _Michel Marcus_, Sep 05 2013
%C The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k. - _Peter Bala_, Jan 31 2022
%C Conjectures: the supercongruences a(p) == -1 - p (mod p^2) and a(2*p) == p - 1 (mod p^2) hold for all primes p >= 3. - _Peter Bala_, Apr 18 2023
%H Seiichi Manyama, <a href="/A008705/b008705.txt">Table of n, a(n) for n = 0..2856</a> (terms 0..256 from N. J. A. Sloane)
%H Morris Newman, <a href="http://dx.doi.org/10.4153/CJM-1958-058-4">Further identities and congruences for the coefficients of modular forms</a>, Canadian J. Math 10 (1958): 577-586. See Table 1, column p=5.
%H Morris Newman, <a href="/A262308/a262308.pdf">Further identities and congruences for the coefficients of modular forms</a> [annotated scanned copy], Canadian J. Math 10 (1958): 577-586. See Table 1, column p=5.
%F a(n) = [x^n] exp(-n*Sum_{k>=1} x^k/(k*(1 - x^k))). - _Ilya Gutkovskiy_, May 30 2018
%e (1-x)^1 = -x + 1, hence a(1) = -1.
%e (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.
%p 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;
%p [seq(C5(i),i=0..30)]; # _N. J. A. Sloane_, Oct 04 2015
%p # second Maple program:
%p b:= proc(n, k) option remember; `if`(n=0, 1, -k*
%p add(numtheory[sigma](j)*b(n-j, k), j=1..n)/n)
%p end:
%p a:= n-> b(n$2):
%p seq(a(n), n=0..35); # _Alois P. Heinz_, Jun 21 2018
%t 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 *)
%o (PARI) a(n) = polcoeff(prod(m = 1, n, (1-x^m)^n), n); \\ _Michel Marcus_, Sep 05 2013
%Y Bisections: A262308, A262309.
%Y Main diagonal of A286354.
%K sign
%O 0,4
%A T. Forbes (anthony.d.forbes(AT)googlemail.com)
%E More terms from _Michel Marcus_, Sep 05 2013
%E a(0)=1 prepended by _N. J. A. Sloane_, Oct 04 2015
|
