A391794 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).

1, 0, 0, 1, 1, 3, 3, 7, 8, 15, 16, 30, 33, 57, 62, 105, 116, 189, 208, 331, 369, 569, 636, 960, 1083, 1593, 1804, 2604, 2967, 4199, 4802, 6684, 7679, 10519, 12118, 16376, 18925, 25243, 29231, 38551, 44733, 58369, 67822, 87658, 101984, 130649, 152120, 193326

Offset

0,6

Comments

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.

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

Links

Alejandro Radisic, Table of n, a(n) for n = 0..500

Formula

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))).

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

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

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

Example

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

Prog

(Python)
def a_list(nmax):
    K = 2 * nmax + 2
    c = [0] * (nmax + 1)
    t = [0] * (nmax + 1)
    t[0] = 1
    for k in range(K + 1):
        s = -1 if (k & 1) else 1
        for j in range(nmax + 1):
            c[j] += s * t[j]
        m1 = k + 2
        if m1 <= nmax:
            for j in range(m1, nmax + 1):
                t[j] += t[j - m1]
        m2 = 2 * (k + 1)
        if m2 <= nmax:
            for j in range(m2, nmax + 1):
                t[j] += t[j - m2]
    return c
print(a_list(47))

Crossrefs

Cf. A002513, A000041.

Sequence in context: A218572 A218573 A117989 * A241642 A086543 A281616

Adjacent sequences: A391791 A391792 A391793 * A391795 A391796 A391797

Keyword

nonn

Author

Alejandro Radisic, Feb 23 2026

Status

approved