A393567 - OEIS

%I #22 Apr 01 2026 22:16:55

%S 1,0,1,-3,8,-20,52,-134,342,-877,2254,-5783,14836,-38076,97712,

%T -250738,643436,-1651170,4237168,-10873260,27902576,-71602589,

%U 183743960,-471517110,1209990193,-3105033096,7968023710,-20447254552,52471005784,-134649198974

%N Coefficients of the formal power series quotient of the '5th-order' mock theta function f_0(q) by the 3rd-order mock theta function f(q).

%C This is the coefficient sequence a(n) defined by A(q) = f_0(q)/f(q), where f_0(q) = Sum_{n>=0} q^(n^2)/(-q;q)_n and f(q) = Sum_{n>=0} q^(n^2)/(-q;q)_n^2.

%C Here (-q;q)_n = Product_{k=1..n} (1 + q^k). Since f(q) has constant term 1, the quotient is well-defined as a formal power series over the rationals.

%D George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Adv. Math. 73 (1989), 242-255.

%D Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355.

%D George N. Watson, The final problem: an account of the mock theta functions, J. London Math. Soc. 11 (1936), 55-80.

%F G.f.: Sum_{n>=0} a(n)*q^n = (Sum_{n>=0} q^(n^2)/(-q;q)_n) / (Sum_{n>=0} q^(n^2)/(-q;q)_n^2).

%F a(0)=1 and for n>=1, a(n) = A053256(n) - Sum_{k=1..n} A000025(k)*a(n-k).

%e A(q) = 1 + q^2 - 3*q^3 + 8*q^4 - 20*q^5 + 52*q^6 - 134*q^7 + ...

%e Hence a(0)=1, a(1)=0, a(2)=1, a(3)=-3, a(4)=8.

%o (Python)

%o from fractions import Fraction

%o def conv(a, b, N):

%o     c = [Fraction(0)] * (N + 1)

%o     for i, ai in enumerate(a):

%o         if ai == 0:

%o             continue

%o         for j, bj in enumerate(b[:N + 1 - i]):

%o             c[i + j] += ai * bj

%o     return c

%o def invfac(k, p, N):

%o     c = [Fraction(0)] * (N + 1)

%o     j = 0

%o     while j * k <= N:

%o         c[j * k] = Fraction(((-1) ** j) * (j + 1 if p == 2 else 1))

%o         j += 1

%o     return c

%o def mock(N, p):

%o     s = [Fraction(0)] * (N + 1)

%o     d = [Fraction(1)] + [Fraction(0)] * N

%o     n = 0

%o     while n * n <= N:

%o         for i, u in enumerate(d[:N + 1 - n * n]):

%o             s[n * n + i] += u

%o         n += 1

%o         d = conv(d, invfac(n, p, N), N)

%o     return s

%o def div(num, den, N):

%o     a = [Fraction(0)] * (N + 1)

%o     for n in range(N + 1):

%o         t = num[n] - sum(den[k] * a[n - k] for k in range(1, n + 1))

%o         a[n] = t / den[0]

%o     return a

%o N = 29

%o print([int(x) for x in div(mock(N, 1), mock(N, 2), N)])

%Y Cf. A000025, A053256.

%K sign

%O 0,4

%A _Joesph Daniel Burke III_, Mar 27 2026