A391794 - OEIS

%I #60 Mar 07 2026 11:16:16

%S 1,0,0,1,1,3,3,7,8,15,16,30,33,57,62,105,116,189,208,331,369,569,636,

%T 960,1083,1593,1804,2604,2967,4199,4802,6684,7679,10519,12118,16376,

%U 18925,25243,29231,38551,44733,58369,67822,87658,101984,130649,152120,193326

%N Coefficients in the expansion of A(q) = lim_{M->inf} Sum_{n=0..2M} (-1)^n / ((q^2;q)_n * (q^2;q^2)_n).

%C Let U(k,q) = 1/((q^2;q)_k * (q^2;q^2)_k). The naive alternating sum Sum_{n>=0} (-1)^n * U(n,q) diverges since U(k,q) -> 1/((q^2;q)_inf * (q^2;q^2)_inf) != 0. The even and odd partial sums converge to different limits differing by 1/((q^2;q)_inf * (q^2;q^2)_inf). A(q) is defined as the even partial-sum limit.

%C All terms appear to be nonnegative (verified for n <= 10000).

%H Alejandro Radisic, <a href="/A391794/b391794.txt">Table of n, a(n) for n = 0..500</a>

%F G.f.: A(q) = lim_{M->inf} Sum_{n=0..2M} (-1)^n U(n,q), where U(k,q) = 1/(Product_{j=0..k-1} (1-q^(j+2)) * Product_{j=0..k-1} (1-q^(2*j+2))).

%F Equivalent: A(q) = 1/((q^2;q)_inf*(q^2;q^2)_inf) + Sum_{m>=0} (U(2m,q) - U(2m+1,q)).

%F Conjectured from n <= 10000: log(a(n)) = Pi*sqrt(n) - (7/4)*log(n) + C + O(n^(-1/2)), C ~ -2.34.

%F Conjectured: a(n) ~ (Pi/4)*b(n)/sqrt(n), where b(n) = A002513(n).

%e A(q) = 1 + q^3 + q^4 + 3*q^5 + 3*q^6 + 7*q^7 + 8*q^8 + 15*q^9 + ...

%o (Python)

%o def a_list(nmax):

%o     K = 2 * nmax + 2

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

%o     t = [0] * (nmax + 1)

%o     t[0] = 1

%o     for k in range(K + 1):

%o         s = -1 if (k & 1) else 1

%o         for j in range(nmax + 1):

%o             c[j] += s * t[j]

%o         m1 = k + 2

%o         if m1 <= nmax:

%o             for j in range(m1, nmax + 1):

%o                 t[j] += t[j - m1]

%o         m2 = 2 * (k + 1)

%o         if m2 <= nmax:

%o             for j in range(m2, nmax + 1):

%o                 t[j] += t[j - m2]

%o     return c

%o print(a_list(47))

%Y Cf. A002513, A000041.

%K nonn

%O 0,6

%A _Alejandro Radisic_, Feb 23 2026