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A184362 G.f.: eta(x) + x*eta'(x).

%I A184362 #17 Feb 11 2025 00:00:57
%S A184362 1,-2,-3,0,0,6,0,8,0,0,0,0,-13,0,0,-16,0,0,0,0,0,0,23,0,0,0,27,0,0,0,
%T A184362 0,0,0,0,0,-36,0,0,0,0,-41,0,0,0,0,0,0,0,0,0,0,52,0,0,0,0,0,58,0,0,0,
%U A184362 0,0,0,0,0,0,0,0,0,-71,0,0,0,0,0,0,-78,0,0,0,0,0,0,0,0,0,0,0,0,0,0,93,0,0,0
%N A184362 G.f.: eta(x) + x*eta'(x).
%C A184362  The formulas specified in this entry use eta(x) to denote Dedekind's eta(q) function without the q^(1/24) factor.
%F A184362 G.f. A(x) satisfies:
%F A184362 (1) [x^n] A(x)/eta(x)^(n+1) = 0 for n>=1.
%F A184362 (2) [x^n] A(x)/eta(x)^n = A109084(n) for n>=0.
%F A184362 (3) [x^n] A(x)/eta(x)^(n+2) = A109085(n) for n>=0.
%F A184362 (4) A(x)/eta(x) = 1 - Sum_{n>=1} sigma(n)*x^n.
%F A184362 (5) A(x) = 1 + Sum_{n>=1} (-1)^n*(n*(3*n-1)/2+1 + (n*(3*n+1)/2+1)*x^n) * x^(n*(3*n-1)/2).
%F A184362 (6) A(x)*eta(x)^2 = Sum_{n>=0} (-1)^n*(2n+1)*(n^2+n+6)/6*x^(n(n+1)/2).
%e A184362 G.f.: A(x) = 1 - 2*x - 3*x^2 + 6*x^5 + 8*x^7 - 13*x^12 - 16*x^15 + 23*x^22 + 27*x^26 - 36*x^35 - 41*x^40 +...
%e A184362 Illustrate the property: [x^n] A(x)/eta(x)^(n+1) = 0
%e A184362 in the table of coefficients of A(x)/eta(x)^(n+1) for n=0..10:
%e A184362 [1, -1, -3, -4, -7, -6, -12, -8, -15, -13, -18,...,-sigma(n),...];
%e A184362 [1,(0), -2, -6, -15, -28, -55, -90, -154, -240, -378,...];
%e A184362 [1, 1,(0), -5, -20, -54, -130, -275, -555, -1050, -1924,...];
%e A184362 [1, 2, 3,(0), -17, -72, -221, -572, -1350, -2958, -6160,...];
%e A184362 [1, 3, 7, 10,(0), -63, -287, -930, -2580, -6475, -15162,...];
%e A184362 [1, 4, 12, 26, 38,(0), -253, -1196, -4059, -11780, -31027,...];
%e A184362 [1, 5, 18, 49, 105, 153,(0), -1062, -5175, -18140, -54544,...];
%e A184362 [1, 6, 25, 80, 210, 442, 646,(0), -4615, -22990, -82671,...];
%e A184362 [1, 7, 33, 120, 363, 924, 1926, 2816,(0), -20570, -104285,...];
%e A184362 [1, 8, 42, 170, 575, 1668, 4161, 8602, 12585,(0), -93538,...];
%e A184362 [1, 9, 52, 231, 858, 2756, 7766, 19071, 39182, 57343,(0),...]; ...
%e A184362 so that the coefficient of x^n in A(x)/eta(x)^(n+1) is zero for n>=1.
%e A184362 Note: the g.f.s of the diagonals in the above table are powers of G(x),
%e A184362 where G(x) = 1/eta(x*G(x)) is the g.f. of A109085.
%e A184362 The g.f. of A184363 equals:
%e A184362 A(x)*eta(x)^2 = 1 - 4*x + 10*x^3 - 21*x^6 + 39*x^10 - 66*x^15 +...+ (-1)^n*(2n+1)*(n^2+n+6)/6*x^(n(n+1)/2) +...
%o A184362 (PARI) {a(n)=polcoeff(eta(x+x*O(x^n)) + x*deriv(eta(x+x*O(x^n))),n)}
%Y A184362 Cf. A184363, A184365, A109084, A109085, A000203.
%K A184362 sign
%O A184362 0,2
%A A184362 _Paul D. Hanna_, Jan 18 2011
%E A184362 Example of g.f. corrected by _Paul D. Hanna_, Jan 18 2011
%E A184362 Name changed slightly by _Paul D. Hanna_, Nov 27 2012