Eric Weisstein's World of Mathematics, <a href="httphttps://mathworld.wolfram.com/QuintupleProductIdentity.html">Quintuple Product Identity</a>.

Exponents of q in the expansion of Sum_{n >= 0} ( q^n * Product_{k = 1..n} (1 - q^(2*k-1)) ) = 1 + q - q^5 - q^8 + q^16 + q^21 - - + + .... - Peter Bala, Dec 03 2020

Exponents of q in the expansion of Product_{n >= 1} (1 - q^(6*n))*(1 + q^(6*n-1))*(1 + q^(6*n-5)) = 1 + q + q^5 + q^8 + q^16 + q^21 + .... - Peter Bala, Dec 09 2020

Exponents of q in the expansion of Product_{n >= 1} (1 - q^n)^2*(1 - q^(4*n))^2 /(1 - q^(2*n)) = 1 - 2*q + 4*q^5 - 5*q^8 + 7*q^16 - + ... (a consequence of the quintuple product identity). The series coefficients are a signed version of A001651. - Peter Bala, Feb 16 2021

Apart from the first two terms, the exponents of q in the expansion of Sum_{n >= 1} q^(3*n+2) * (Product_{k = 2..n} 1 - q^(2*k-1)) = q^5 + q^8 - q^16 - q^21 + + - - ... (in Andrews, equation 8, replace q with q^2 and set x = q).

Exponents of q^2 in the expansion in powers of q^2 of Sum_{n >= 0} q^n / (Product_{k = 1..n+1 } 1 + q^(2*k-1)) = 1 + (q^2)^1 - (q^2)^5 - (q^2)^8 + (q^2)^16 + (q^2)^21 - - + + ... (Chen, equation 22). (End)

Exponents in the expansion of Product_{n >= 1} (1 - q^(2*n-1))^2 * (1 - q^(4*n-2)) * (1 - q^(4*n))^3 = 1 - 2*q + 4*q^5 - 5*q^8 + 7*q^16 - 8*q^21 + 10*q^33 - 11*q^40 + - ... (the unsigned coefficients are the positive integers not divisible by 3). - Peter Bala, Dec 25 2024

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Exponents of q in the expansion of Sum_{n >= 0} ( q^n * Product_{k = 1..n} (1 - q^(2*k-1)) ) = 1 + q - q^5 - q^8 + q^16 + q^21 - - + + .... - Peter Bala, Dec 03 2020

Exponents of q in the expansion of Product_{n >= 1} (1 - q^(6*n))*(1 + q^(6*n-1))*(1 + q^(6*n-5)) = 1 + q + q^5 + q^8 + q^16 + q^21 + .... - Peter Bala, Dec 09 2020

Exponents of q in the expansion of Product_{n >= 1} (1 - q^n)^2*(1 - q^(4*n))^2 /(1 - q^(2*n)) = 1 - 2*q + 4*q^5 - 5*q^8 + 7*q^16 - + ... (a consequence of the quintuple product identity). The series coefficients are a signed version of A001651. - Peter Bala, Feb 16 2021

Apart from the first two terms, the exponents of q in the expansion of Sum_{n >= 1} q^(3*n+2) * (Product_{k = 2..n} 1 - q^(2*k-1)) = q^5 + q^8 - q^16 - q^21 + + - - ... (in Andrews, equation 8, replace q with q^2 and set x = q).

Exponents of q^2 in the expansion in powers of q^2 of Sum_{n >= 0} q^n / (Product_{k = 1..n+1 } 1 + q^(2*k-1)) = 1 + (q^2)^1 - (q^2)^5 - (q^2)^8 + (q^2)^16 + (q^2)^21 - - + + ... (Chen, equation 22). (End)

Exponents in the expansion of Product_{n >= 1} (1 - q^(2*n-1))^2 * (1 - q^(4*n-2)) * (1 - q^(4*n))^3 = 1 - 2*q + 4*q^5 - 5*q^8 + 7*q^16 - 8*q^21 + 10*q^33 - 11*q^40 + - ... (the unsigned coefficients are the positive integers not divisible by 3). - Peter Bala, Dec 25 2024

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Integer quotients of (x*(x+1)*(x+2))/(x+(x+1)+(x+2)) for positive values of x which are given by A032766. - Patrick De Geest, Dec 13 2024

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Integer solutions quotients of (x*(x+1)*(x+2))/(x+(x+1)+(x+2)). Values for positive values of x which are given by A032766. - Patrick De Geest, Dec 13 2024

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