A379212 Expansion of Sum_{n >= 0} q^(n*(n+1)) * Product_{k >= 2*n+2} 1 - q^k.

1, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0

Offset

0

Comments

Compare with the expansion Sum_{n >= 0} q^(n*(n+1)) * Product_{k >= 2*n+1} 1 - q^k = 1 - q - q^8 + q^13 + q^17 - q^24 - q^45 + + - - ..., where the sequence of exponents [0, 1, 8, 13, 17, 24, 45, ...] is A204221.

References

George E. Andrews, Richard Askey, and Ranjan Roy, Special Functions, Cambridge University Press, 1999.

Links

Table of n, a(n) for n=0..100.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

G.f.: A(q) = Product_{n >= 0} (1 - q^(20*n+3))*(1 - q^(20*n+4))*(1 - q^(20*n+7))*(1 - q^(10*n+10))*(1 - q^(20*n+13))*(1 - q^(20*n+16))*(1 - q^(20*n+17)) = 1 - q^3 - q^4 + q^11 + q^19 - q^32 - q^35 + + - - .... See Andrews et al., p. 591, Exercise 6(d).

q^4 * A(q^15) = q^4 - q^49 - q^64 + q^169 + q^289 - q^484 - q^529 + + - - .... The exponents are given by A379211(n)^2.

|a(n)| is the characteristic function of A379210.

Maple

series(add(q^(n*(n+1)) * mul(1 - q^k, k = 2*n+2..100), n = 0..10), q, 101):
seq(coeftayl(%, q = 0, n), n = 0..100);

Crossrefs

Cf. A204220, A204221, A379210, A379211.

Sequence in context: A126999 A306862 A363801 * A347283 A214293 A323096

Adjacent sequences: A379209 A379210 A379211 * A379213 A379214 A379215

Keyword

sign,easy,changed

Author

Peter Bala, Dec 18 2024

Status

approved