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 A281452 Expansion of f(x, x) * f(x^5, x^13) in powers of x where f(, ) is Ramanujan's general theta function. 3
 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 (list; graph; refs; listen; history; text; internal format)
 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: A332662 A029303 A281640 * A341023 A241067 A130457 Adjacent sequences:  A281449 A281450 A281451 * A281453 A281454 A281455 KEYWORD nonn AUTHOR Michael Somos, Jan 26 2017 STATUS approved

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Last modified November 30 20:17 EST 2021. Contains 349425 sequences. (Running on oeis4.)