A281452 Expansion of f(x, x) * f(x^5, x^13) in powers of x where f(, ) is Ramanujan's general theta function.

1, 2, 0, 0, 2, 1, 2, 0, 0, 4, 0, 0, 0, 1, 4, 0, 2, 2, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 1, 4, 2, 0, 2, 0, 0, 0, 2, 2, 2, 0, 0, 2, 0, 0, 3, 2, 0, 0, 2, 4, 0, 0, 0, 4, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 4, 0, 0, 0, 0, 5, 2, 0, 0, 2, 0, 0, 0, 4, 2, 0, 2, 2, 0, 0, 0, 2, 2

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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

Euler transform of a period 36 sequence.

G.f.: (Sum_{k in Z} x^k^2) * (Sum_{k in Z} x^(9*k^2 + 4*k)).

G.f.: Product_{k>0} (1 + x^(2*k-1))^2 * (1 - x^(2*k)) * (1 - x^(18*k-13)) * (1 - x^(18*k-5)) * (1 - x^(18*k)).

a(n) = A122865(3*n + 1) = A122856(6*n + 2) = A258278(6*n + 2). a(n) = - A256269(9^n + 4). 4 * a(n) = A004018(9*n + 4).

a(n) = b(9*n + 4) with b = A002654, A035154, A113446, A122864, A125061, A129448, A138950, A163746, A256276, A258228, A258256.

2 * a(n) = b(9*n + 4) = with b = A105673, A105673, A122857, A258034, A259761. -2 * a(n) = b(9*n + 4) with b = A138949, A256280, A258292.

a(4*n) = A281453(n). a(8*n + 6) = 2 * A281490(n). a(16*n + 12) = A281451(n).

a(32*n + 4) = 2 * A281492(n). a(64*n + 28) = A281452(n). a(128*n + 60) = 2 * A281491(n).

Example

G.f. = 1 + 2*x + 2*x^4 + x^5 + 2*x^6 + 4*x^9 + x^13 + 4*x^14 + 2*x^16 + ...
G.f. = q^4 + 2*q^13 + 2*q^40 + q^49 + 2*q^58 + 4*q^85 + q^121 + 4*q^130 + ...

Mathematica

a[ n_] := If[ n < 0, 0, DivisorSum[ 9 n + 4, KroneckerSymbol[ -4, #] &]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x^5, x^18] QPochhammer[ -x^13, x^18] QPochhammer[ x^18], {x, 0, n}];
a[ n_] := If[ n < 0, 0, Times @@ (Which[ # < 3, 1, Mod[#, 4] == 1, #2 + 1, True, (1 + (-1)^#2) / 2] & @@@ FactorInteger[ 9 n + 4])];

Prog

(PARI) {a(n) = if( n<0, 0, sumdiv(9*n + 4, d, (d%4==1) - (d%4==3)))};
(PARI) {a(n) = if( n<0, 0, my(m = 9*n + 4, k, s); forstep(j=0, sqrtint(m), 3, if( issquare(m - j^2, &k) && (k%9 == 2 || k%9 == 7), s+=(j>0)+1)); s)};
(PARI) {a(n) = if( n<0, 0, my(A, p, e); A = factor(9*n + 4); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==2, 1, p%4==1, e+1, 1-e%2)))};

Crossrefs

Cf. A002654, A004018, A035154, A105673, A113446, A122856, A122857, A122864, A122865.

Cf. A125061, A129448, A138949, A138950, A163746, A256269, A256276, A256280, A258034.

Cf. A258228, A258256, A258278, A258292, A259761.

Cf. A281451, A281453, A281490, A281491, A281492.

Sequence in context: A361163 A361500 A281640 * A341023 A241067 A130457

Adjacent sequences: A281449 A281450 A281451 * A281453 A281454 A281455

Keyword

nonn

Author

Michael Somos, Jan 26 2017

Status

approved