A010822 - OEIS

%I #28 Dec 19 2018 09:03:53

%S 1,-15,90,-245,105,1107,-2485,195,4860,-2420,-3990,-8190,19695,13755,

%T -38475,3990,-9750,34020,43015,-46605,-13860,-127385,106485,165240,

%U -79275,-16380,-92340,-35840,-151995,188550,315783,90090,-271215,-307485,20475,-505440,915385,209340,-284130,337645,-294225,269325,-1707970,-70305,1297620,574210,492765,251370,-847245,-1102725,438129,-1416190,641445,0

%N Expansion of Product_{k>=1} (1 - x^k)^15.

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

%D Van der Blij, F. "The function tau(n) of S. Ramanujan (an expository lecture)." Math. Student 18 (1950): 83-99. See page 85.

%H Seiichi Manyama, <a href="/A010822/b010822.txt">Table of n, a(n) for n = 0..10000</a>

%H M. Boylan, <a href="http://dx.doi.org/10.1016/S0022-314X(02)00037-9">Exceptional congruences for the coefficients of certain eta-product newforms</a>, J. Number Theory 98 (2003), no. 2, 377-389.

%H <a href="/index/Pro#1mxtok">Index entries for expansions of Product_{k >= 1} (1-x^k)^m</a>

%F a(0) = 1, a(n) = -(15/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Mar 27 2017

%F G.f.: exp(-15*Sum_{k>=1} x^k/(k*(1 - x^k))). - _Ilya Gutkovskiy_, Feb 05 2018

%e 1 - 15*x + 90*x^2 - 245*x^3 + 105*x^4 + 1107*x^5 - 2485*x^6 + 195*x^7 + ...

%t a[0] = 1; a[n_] := a[n] = -15/n Sum[DivisorSigma[1, k] a[n-k], {k, 1, n}];

%t Table[a[n], {n, 0, 53}] (* _Jean-François Alcover_, Dec 19 2018, after _Seiichi Manyama_ *)

%Y See A322043 for the coefficients that are zero.

%Y Cf. also A000594.

%K sign

%O 0,2

%A _N. J. A. Sloane_.