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A133100 Expansion of f(x, x^4) in powers of x where f(, ) is Ramanujan's general theta function.

%I A133100 #29 Jan 13 2024 03:40:25
%S A133100 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 A133100 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 A133100 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 A133100 Expansion of f(x, x^4) in powers of x where f(, ) is Ramanujan's general theta function.
%H A133100 Seiichi Manyama, <a href="/A133100/b133100.txt">Table of n, a(n) for n = 0..1000</a>
%H A133100 <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>.
%F A133100 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 A133100 The characteristic function of A085787 generalized heptagonal numbers.
%F A133100 Euler transform of period 10 sequence [1, -1, 0, 1, -1, 1, 0, -1, 1, -1, ...].
%F A133100 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 A133100 a(n) = |A113429(n)|. a(3*n + 2) = 0.
%F A133100 Sum_{k=1..n} a(k) ~ 2 * sqrt(2/5) * sqrt(n). - _Amiram Eldar_, Jan 13 2024
%e A133100 G.f. = 1 + x + x^4 + x^7 + x^13 + x^18 + x^27 + x^34 + x^46 + x^55 + x^70 + ...
%e A133100 G.f. = q^9 + q^49 + q^169 + q^289 + q^529 + q^729 + q^1089 + q^1369 + q^1849 + ...
%t A133100 a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^5] QPochhammer[ -x^4, x^5] QPochhammer[ x^5], {x, 0, n}]; (* _Michael Somos_, Oct 31 2015 *)
%t A133100 a[ n_] := SquaresR[ 1, 40 n + 9] / 2; (* _Michael Somos_, Jan 30 2017 *)
%t A133100 a[ n_] := If[n < 0, 0, Boole @ IntegerQ @ Sqrt @ (40 n + 9)]; (* _Michael Somos_, Jan 30 2017 *)
%o A133100 (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 A133100 (PARI) {a(n) = issquare( 40*n + 9)};
%Y A133100 Cf. f(x, x^k) for k=2..11: A080995, A010054, A133100, A089801, A274179, A214263, A281814, A205988, A281815, A186742.
%Y A133100 Cf. A085787, A113429.
%K A133100 nonn
%O A133100 0,1
%A A133100 _Michael Somos_, Sep 11 2007