A281451 -id:A281451 - cp's OEIS frontend

A281490 Expansion of f(x, x^3) * f(x, x^8) in powers of x where f(, ) is Ramanujan's general theta function.

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

Offset: 0

Michael Somos, Jan 29 2017

nonn

			G.f. = 1 + 2*x + x^2 + x^3 + x^4 + x^6 + x^7 + x^8 + x^9 + x^10 + 3*x^11 + ...
G.f. = q^29 + 2*q^65 + q^101 + q^137 + q^173 + q^245 + q^281 + q^317 + q^353 + ...

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

Cf. A010054, A281451, A281814.

a[ n_] := SeriesCoefficient[ (1/2) x^(-1/8) EllipticTheta[ 2, 0, x^(1/2)] QPochhammer[ -x, x^9] QPochhammer[ -x^8, x^9] QPochhammer[ x^9], {x, 0, n}];

{a(n) = if( n<0, 0, sumdiv(36*n + 29, d, kronecker(-4, d)) / 2)};

{a(n) = if( n<0, 0, my(A, p, e); n = 36*n + 29; A = factor(n); prod(k=1, matsize(A) [1], [p, e] = A[k, ]; if(p%4==1, e+1, 1-e%2)) / 2)};

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
G.f.: (Sum_{k>0} x^(k*(k - 1)/2)) * (Sum_{k in Z} x^(k*(9*k + 7)/2)).
G.f.: Product_{k>0} (1 - x^(2*k)) / (1 - x^(2*k-1)) * (1 + x^(9*k-8)) * (1 + x^(9*k-1)) * (1 - x^(9*k)).
Convolution of sequences A010054 and A281814.
2 * a(n) = A281451(32*n + 25).

A281491 Expansion of f(x, x^3) * f(x^2, x^7) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Jan 29 2017

nonn

			G.f. = 1 + x + x^2 + 2*x^3 + x^5 + x^6 + x^7 + 2*x^8 + 2*x^10 + x^12 + ...
G.f. = q^17 + q^53 + q^89 + 2*q^125 + q^197 + q^233 + q^269 + 2*q^305 + ...

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

Cf. A281451.

a[ n_] := SeriesCoefficient[ (1/2) x^(-1/8) EllipticTheta[ 2, 0, x^(1/2)] QPochhammer[ -x^2, x^9] QPochhammer[ -x^7, x^9] QPochhammer[ x^9], {x, 0, n}];

{a(n) = if( n<0, 0, sumdiv(36*n + 17, d, kronecker(-4, d)) / 2)};

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
Euler transform of period 18 sequence [1, 0, 1, -2, 1, -1, 2, -1, 0, -1, 2, -1, 1, -2, 1, 0, 1, -2, ...].
G.f.: (Sum_{k>0} x^(k*(k - 1)/2)) * (Sum_{k in Z} x^(k*(9*k + 5)/2)).
G.f.: Product_{k>0} (1 - x^(2*k)) / (1 - x^(2*k-1)) * (1 + x^(9*k-7)) * (1 + x^(9*k-2)) * (1 - x^(9*k)).
2 * a(n) = A281451(8*n + 3).

A281492 Expansion of f(x, x^3) * f(x^4, x^5) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Jan 29 2017

nonn

			G.f. = 1 + x + x^3 + x^4 + 2*x^5 + 2*x^6 + x^7 + x^8 + 2*x^10 + x^11 + ...
G.f. = q^5 + q^41 + q^113 + q^149 + 2*q^185 + 2*q^221 + q^257 + q^293 + ...

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

Cf. A281451.

a[ n_] := SeriesCoefficient[ (1/2) x^(-1/8) EllipticTheta[ 2, 0, x^(1/2)] QPochhammer[ -x^4, x^9] QPochhammer[ -x^5, x^9] QPochhammer[ x^9], {x, 0, n}];

{a(n) = if( n<0, 0, sumdiv(36*n + 5, d, kronecker(-4, d)) / 2)};

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 period 18 sequence [1, -1, 1, 0, 2, -1, 1, -2, 0, -2, 1, -1, 2, 0, 1, -1, 1, -2, ...].
G.f.: (Sum_{k>0} x^(k*(k - 1)/2)) * (Sum_{k in Z} x^(k*(9*k + 1)/2)).
G.f.: Product_{k>0} (1 - x^(2*k)) / (1 - x^(2*k-1)) * (1 + x^(9*k-5)) * (1 + x^(9*k-4)) * (1 - x^(9*k)).
2 * a(n) = A281451(128*n + 17).

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

Original entry on oeis.org

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

Michael Somos, Jan 26 2017

nonn

			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 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A002654, A004018, A019670, 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.

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])];

{a(n) = if( n<0, 0, sumdiv(9*n + 4, d, (d%4==1) - (d%4==3)))};

{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)};

{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)))};

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).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 = 1.0471975... (A019670). - Amiram Eldar, Jan 20 2025

A281453 Expansion of f(x, x) * f(x^7, x^11) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

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, 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, 2, 0, 0, 2, 0, 0, 0, 3, 4, 0, 0, 2, 0, 4, 0, 0, 2, 0, 0

Offset: 0

Michael Somos, Jan 26 2017

nonn

			G.f. = 1 + 2*x + 2*x^4 + x^7 + 2*x^8 + 2*x^9 + 3*x^11 + 2*x^12 + ...
G.f. = q + 2*q^10 + 2*q^37 + q^64 + 2*q^73 + 2*q^82 + 3*q^100 + ...

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

Cf. A000122, A002654, A004018, A035154, A091400, A105673, A113446, A122856, A122857.
Cf. A122864, A122865, A125061, A129448, A138949, A138950, A163746, A205808, A256276.
Cf. A256280, A258034, A258228, A258256, A258278, A258292, A259761.
Cf. A281451, A281452, A281490, A281491, A281492.

a[ n_] := If[ n < 0, 0, DivisorSum[ 9 n + 1, KroneckerSymbol[ -4, #] &]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x^7, x^18] QPochhammer[ -x^11, x^18] QPochhammer[ x^18], {x, 0, n}];
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])];

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

{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)};

{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)))};

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 + 2*k)).
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)).
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.
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).
a(n) = A122865(3*n) = A122856(6*n) = A258278(6*n) = a(64*n + 7). a(n) = -A256269(9*n + 1).
a(n) = b(9*n + 1) where b = A002654, A035154, A091400, A113446, A122864, A125061, A129448, A138950, A163746, A256276, A258228, A258256.
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).
Convolution of A000122 and A205808.

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A281490 Expansion of f(x, x^3) * f(x, x^8) in powers of x where f(, ) is Ramanujan's general theta function.

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A281491 Expansion of f(x, x^3) * f(x^2, x^7) in powers of x where f(, ) is Ramanujan's general theta function.

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A281492 Expansion of f(x, x^3) * f(x^4, x^5) in powers of x where f(, ) is Ramanujan's general theta function.

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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.

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A281453 Expansion of f(x, x) * f(x^7, x^11) in powers of x where f(, ) is Ramanujan's general theta function.

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