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Expansion of f(x, x^4) in powers of x where f(, ) is Ramanujan's general theta function.
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%I #29 Jan 13 2024 03:40:25

%S 1,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,

%T 1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,

%U 0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0

%N Expansion of f(x, x^4) in powers of x where f(, ) is Ramanujan's general theta function.

%H Seiichi Manyama, <a href="/A133100/b133100.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>.

%F f(x,x^m) = 1 + Sum_{k=1..oo} x^((m+1)*k*(k-1)/2) (x^k + x^(m*k)). - _N. J. A. Sloane_, Jan 30 2017

%F The characteristic function of A085787 generalized heptagonal numbers.

%F Euler transform of period 10 sequence [1, -1, 0, 1, -1, 1, 0, -1, 1, -1, ...].

%F G.f.: Prod_{k>0} (1 - x^(5*k)) * (1 + x^(5*k - 1)) * (1 + x^(5*k - 4)) = Sum_{k in Z} x^((5*k^2 + 3*k) / 2).

%F a(n) = |A113429(n)|. a(3*n + 2) = 0.

%F Sum_{k=1..n} a(k) ~ 2 * sqrt(2/5) * sqrt(n). - _Amiram Eldar_, Jan 13 2024

%e G.f. = 1 + x + x^4 + x^7 + x^13 + x^18 + x^27 + x^34 + x^46 + x^55 + x^70 + ...

%e G.f. = q^9 + q^49 + q^169 + q^289 + q^529 + q^729 + q^1089 + q^1369 + q^1849 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^5] QPochhammer[ -x^4, x^5] QPochhammer[ x^5], {x, 0, n}]; (* _Michael Somos_, Oct 31 2015 *)

%t a[ n_] := SquaresR[ 1, 40 n + 9] / 2; (* _Michael Somos_, Jan 30 2017 *)

%t a[ n_] := If[n < 0, 0, Boole @ IntegerQ @ Sqrt @ (40 n + 9)]; (* _Michael Somos_, Jan 30 2017 *)

%o (PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1,n, 1 + x^k*[-1, 1, 0, 0, 1][k%5 + 1], 1 + x * O(x^n)), n))};

%o (PARI) {a(n) = issquare( 40*n + 9)};

%Y Cf. f(x, x^k) for k=2..11: A080995, A010054, A133100, A089801, A274179, A214263, A281814, A205988, A281815, A186742.

%Y Cf. A085787, A113429.

%K nonn

%O 0,1

%A _Michael Somos_, Sep 11 2007