%I #10 Mar 12 2021 22:24:48

%S 1,2,0,0,2,0,0,1,2,2,0,3,2,0,0,2,4,0,0,0,2,0,0,2,0,2,0,2,0,0,0,0,3,2,

%T 0,0,6,0,0,0,1,4,0,2,2,0,0,2,2,4,0,0,0,0,0,0,4,2,0,0,2,0,0,0,2,2,0,0,

%U 2,0,0,2,0,0,0,3,4,0,0,2,0,4,0,0,2,0,0

%N Expansion of f(x, x) * f(x^7, x^11) in powers of x where f(, ) is Ramanujan's general theta function.

%H G. C. Greubel, <a href="/A281453/b281453.txt">Table of n, a(n) for n = 0..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

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

%F Euler transform of a period 36 sequence.

%F G.f.: (Sum_{k in Z} x^k^2) * (Sum_{k in Z} x^(9*k^2 + 2*k)).

%F G.f.: Product_{k>0} (1 + x^(2*k-1))^2 * (1 - x^(2*k)) * (1 + x^(18*k-11)) * (1 + x^(18*k-7)) * (1 - x^(18*k)).

%F a(4*n + 2) = a(8*n + 5) = a(16*n + 3) = a(32*n + 31) = a(64*n + 55) = a(128*n + 39) = 0.

%F a(4*n + 3) = A281451(n). a(8*n + 1) = 2 * A281492(n). a(16*n + 7) = A281452(n). a(32*n + 15) = 2 * A281491(n). a(128*n + 103) = 2 * A281490(n).

%F a(n) = A122865(3*n) = A122856(6*n) = A258278(6*n) = a(64*n + 7). a(n) = -A256269(9*n + 1).

%F a(n) = b(9*n + 1) where b = A002654, A035154, A091400, A113446, A122864, A125061, A129448, A138950, A163746, A256276, A258228, A258256.

%F 2 * a(n) = b(9*n + 1) where b = A105673, A122857, A258034, A259761. 2 * a(n) = - b(9*n+1) where b = A138949, A256280, A258292. 4 * a(n) = A004018(9*n + 1).

%F Convolution of A000122 and A205808.

%e G.f. = 1 + 2*x + 2*x^4 + x^7 + 2*x^8 + 2*x^9 + 3*x^11 + 2*x^12 + ...

%e G.f. = q + 2*q^10 + 2*q^37 + q^64 + 2*q^73 + 2*q^82 + 3*q^100 + ...

%t a[ n_] := If[ n < 0, 0, DivisorSum[ 9 n + 1, KroneckerSymbol[ -4, #] &]];

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x^7, x^18] QPochhammer[ -x^11, x^18] QPochhammer[ x^18], {x, 0, n}];

%t a[ n_] := If[ n < 0, 0, Times @@ (Which[# < 3, 1, # == 3, Mod[#2, 2] 2 + 1, Mod[#, 4] == 1, #2 + 1, True, (1 + (-1)^#2) / 2] & @@@ FactorInteger[ 9 n + 1])];

%o (PARI) {a(n) = if( n<0, 0, sumdiv(9*n + 1, d, kronecker(-4, d)))};

%o (PARI) {a(n) = if( n<0, 0, my(m = 9*n + 1, k, s); forstep(j=0, sqrtint(m), 3, if( issquare(m - j^2, &k) && (k%9 == 1 || k%9 == 8), s+=(j>0)+1)); s)};

%o (PARI) {a(n) = if( n<0, 0, my(A, p, e); A = factor(9*n + 1); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==2, 1, p==3, -2*(-1)^e, p%4==1, e+1, 1-e%2)))};

%Y Cf. A000122, A002654, A004018, A035154, A091400, A105673, A113446, A122856, A122857.

%Y Cf. A122864, A122865, A125061, A129448, A138949, A138950, A163746, A205808, A256276.

%Y Cf. A256280, A258034, A258228, A258256, A258278, A258292, A259761.

%Y Cf. A281451, A281452, A281490, A281491, A281492.

%K nonn

%O 0,2

%A _Michael Somos_, Jan 26 2017