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A281451 Expansion of x * f(x, x) * f(x, x^17) in powers of x where f(, ) is Ramanujan's general theta function.

%I A281451 #27 Feb 16 2025 08:33:39
%S A281451 1,3,2,0,2,2,0,0,0,2,2,0,0,0,0,0,2,3,2,0,1,4,0,0,2,2,4,0,0,2,0,0,0,2,
%T A281451 0,0,4,2,0,0,0,0,2,0,0,2,0,0,0,2,2,0,1,4,0,0,4,1,2,0,0,4,0,0,2,2,4,0,
%U A281451 2,2,0,0,0,2,0,0,0,2,0,0,0,4,4,0,2,0,0
%N A281451 Expansion of x * f(x, x) * f(x, x^17) in powers of x where f(, ) is Ramanujan's general theta function.
%H A281451 G. C. Greubel, <a href="/A281451/b281451.txt">Table of n, a(n) for n = 1..1000</a>
%H A281451 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A281451 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A281451 f(x,x^m) = 1 + Sum_{k>=1} x^((m+1)*k*(k-1)/2) (x^k + x^(m*k)). - _N. J. A. Sloane_, Jan 30 2017
%F A281451 Euler transform of a period 36 sequence.
%F A281451 G.f.: x * (Sum_{k in Z} x^k^2) * (Sum_{k in Z} x^(9*k^2 + 8*k)).
%F A281451 G.f.: x * Product_{k>0} (1 + x^(2*k-1))^2 * (1 - x^(2*k)) * (1 + x^(18*k-17)) * (1 + x^(18*k-1)) * (1 - x^(18*k)).
%F A281451 a(4*n) = a(8*n + 7) = a(16*n + 13) = a(32*n + 9) = a(49*n + 7) = a(98*n + 14) = 0.
%F A281451 a(4*n + 1) = A281452(n).  a(8*n + 3) = 2 * A281491(n).  A(16*n + 1) = A281453(n).
%F A281451 a(32*n + 25) = 2 * A281490(n). a(64*n + 49) = a(n). a(128*n + 17) = 2 * A281492(n).
%F A281451 a(n) = A122865(3*n + 2). a(n) = A122856(6*n + 4) = A258278(6*n + 4).
%F A281451 a(n) = b(9*n + 7) where b = A002654, A035154, A113446, A122864, A125061, A129448, A138950, A163746, A256276, A258228, A258256.
%F A281451 2 * a(n) = b(9*n + 7) where b = A105673, A122857, A258034, A259761.  -2 * a(n) = b(9*n + 7) where b = A138949, A256280, A258292.
%F A281451 a(n) = - A256269(9*n + 7). 4 * a(n) = A004018(9*n + 7).
%F A281451 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 = 1.0471975... (A019670). - _Amiram Eldar_, Jan 20 2025
%e A281451 G.f. = x + 3*x^2 + 2*x^3 + 2*x^5 + 2*x^6 + 2*x^10 + 2*x^11 + 2*x^17 + ...
%e A281451 G.f. = q^16 + 3*q^25 + 2*q^34 + 2*q^52 + 2*q^61 + 2*q^97 + 2*q^106 + ...
%t A281451 a[ n_] := If[ n < 0, 0, DivisorSum[ 9 n + 7, KroneckerSymbol[ -4, #] &]];
%t A281451 a[ n_] := If[ n < 0, 0, Times @@ (Which[# < 3, 1, Mod[#, 4] == 1, #2 + 1, True, (1 + (-1)^#2) / 2] & @@@ FactorInteger[ 9 n + 7])];
%t A281451 a[ n_] := SeriesCoefficient[ x EllipticTheta[ 3, 0, x] QPochhammer[ -x, x^18] QPochhammer[ -x^17, x^18] QPochhammer[ x^18], {x, 0, n}];
%o A281451 (PARI) {a(n) = if( n<0, 0, sumdiv(9*n + 7, d, (d%4==1) - (d%4==3)))};
%o A281451 (PARI) {a(n) = if( n<0, 0, my(A, p, e); A = factor(9*n + 7); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==2, 1, p%4==1, e+1, 1-e%2)))};
%o A281451 (PARI) {a(n) = if( n<0, 0, my(m = 9*n + 7, k, s); forstep(j=0, sqrtint(m), 3, if( issquare(m - j^2, &k) && (k%9 == 4 || k%9 == 5), s+=(j>0)+1)); s)};
%Y A281451 Cf. A002654, A004018, A019670, A035154, A105673, A113446, A122856, A122857, A122864, A122865.
%Y A281451 Cf. A125061, A129448, A138949, A138950, A163746, A256269, A256276, A256280, A258034.
%Y A281451 Cf. A258228, A258256, A258278, A258292, A259761.
%Y A281451 Cf. A281452, A281453, A281490, A281491, A281492.
%K A281451 nonn
%O A281451 1,2
%A A281451 _Michael Somos_, Jan 23 2017