A214264 -id:A214264 - cp's OEIS frontend

A282893 The difference between the number of partitions of 2n into odd parts (A000009) and the number of partitions of 2n into even parts (A035363).

0, 0, 0, 1, 1, 3, 4, 7, 10, 16, 22, 33, 45, 64, 87, 120, 159, 215, 283, 374, 486, 634, 814, 1049, 1335, 1700, 2146, 2708, 3390, 4243, 5276, 6552, 8095, 9989, 12266, 15044, 18375, 22409, 27235, 33049, 39974, 48281, 58148, 69923, 83871, 100452, 120027, 143214, 170515, 202731, 240567, 285073, 337195

Offset: 0

Robert G. Wilson v, Feb 24 2017

nonn

The even bisection of A282892. The other bisection is A078408.

			G.f. = x^3 + x^4 + 3*x^5 + 4*x^6 + 7*x^7 + 10*x^8 + 16*x^9 + 22*x^10 + 33*x^11 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000009, A000041, A035294, A035363, A078408, A214264, A282892.

with(numtheory):
b:= proc(n, t) option remember; `if`(n=0, 1, add(add(`if`(
      (d+t)::odd, d, 0), d=divisors(j))*b(n-j, t), j=1..n)/n)
    end:
a:= n-> b(2*n, 0) -b(2*n, 1):
seq(a(n), n=0..80);  # Alois P. Heinz, Feb 24 2017

f[n_] := Length[ IntegerPartitions[n, All, 2Range[n] -1]] - Length[ IntegerPartitions[n, All, 2 Range[n]]]; Array[ f[2#] &, 52]
a[ n_] := SeriesCoefficient[ Sum[ Sign @ SquaresR[1, 16 k + 1] x^k, {k, n}] / QPochhammer[x], {x, 0, n}]; (* Michael Somos, Feb 24 2017 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( sum(k=1, n, issquare(16*k + 1)*x^k, A) / eta(x + A), n))}; /* Michael Somos, Feb 24 2017 */

a(n) = A282892(2n).
Expansion of (f(x^3, x^5) - 1) / f(-x, -x^2) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Feb 24 2017
a(n) = A035294(n) - A000041(n). - Michael Somos, Feb 24 2017

A244465 Expansion of f(-x^3, -x^5) in powers of x where f() is Ramanujan's two-variable theta function.

Original entry on oeis.org

1, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Michael Somos, Jun 28 2014

sign

			G.f. = 1 - x^3 - x^5 + x^14 + x^18 - x^33 - x^39 + x^60 + x^68 - x^95 + ...
G.f. = q - q^49 - q^81 + q^225 + q^289 - q^529 - q^625 + q^961 + q^1089 + ...

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. A214264.

a[ n_] := SeriesCoefficient[ QPochhammer[ x^3, x^8] QPochhammer[ x^5, x^8] QPochhammer[ x^8], {x, 0, n}];

PARI
```
{a(n) = issquare( 16*n + 1) * (-1)^n};
```

Euler transform of period 8 sequence [ 0, 0, -1, 0, -1, 0, 0, -1, ...].
G.f.: f(-x^3, -x^5) = Sum_{k in Z} (-1)^k * x^(4*k^2 - k).
a(n) = (-1)^n * A214264(n).

A246862 Expansion of phi(x) * f(x^3, x^5) in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, 0, 1, 4, 1, 2, 2, 0, 4, 0, 0, 2, 0, 3, 2, 2, 0, 3, 4, 0, 2, 2, 2, 0, 2, 0, 2, 2, 0, 4, 0, 0, 1, 4, 0, 2, 2, 0, 5, 2, 2, 2, 4, 0, 0, 0, 0, 2, 4, 2, 0, 2, 0, 4, 2, 0, 0, 2, 0, 1, 2, 0, 2, 6, 0, 0, 4, 1, 8, 0, 0, 2, 0, 0, 2, 2, 2, 2, 0, 0, 2, 4, 0, 4, 2, 2

Offset: 0

Michael Somos, Sep 05 2014

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + 2*x + x^3 + 4*x^4 + x^5 + 2*x^6 + 2*x^7 + 4*x^9 + 2*x^12 + ...
G.f. = q + 2*q^17 + q^49 + 4*q^65 + q^81 + 2*q^97 + 2*q^113 + 4*q^145 + ...

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, A008441, A113407, A116604, A125079, A129447, A134343, A138741, A214264, A226192, A246863.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x^3, x^8] QPochhammer[ -x^5, x^8] QPochhammer[ x^8], {x, 0, n}];

{a(n) = if( n<0, 0, issquare(16 * n + 1) + 2 * sum(i=1, sqrtint(n), issquare(16 * (n - i^2) + 1)))};

Euler transform of period 16 sequence [ 2, -3, 3, -1, 3, -4, 2, -2, 2, -4, 3, -1, 3, -3, 2, -2, ...].
Convolution of A000122 and A214264.
a(9*n + 2) = a(9*n + 8) = 0. a(9*n + 5) = A246863(n).
a(n) = A113407(2*n) = A226192(2*n) = A008441(4*n) = A134343(4*n) = A116604(8*n) = A125079(8*n) = A129447(8*n) = A138741(8*n).

A214302 Expansion of f(-x^2, -x^4) * f(x^3, x^5) in powers of x where f(,) is Ramanujan's two-variable theta function.

Original entry on oeis.org

1, 0, -1, 1, -1, 0, 0, -2, 0, -1, 1, 0, 0, 1, 2, 1, -1, 1, 0, 1, -1, 0, -1, 0, 0, 0, 0, -1, 2, -1, -1, 0, 1, 0, 0, -2, 0, -1, -1, 1, 0, -1, -1, 0, 0, 0, 0, 2, -1, 2, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, -2, -1, 0, 1, 0, 1, -1, 0, 0, -1, -1, 1, -1, 0, 1, 1

Offset: 0

Michael Somos, Jul 11 2012

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

a(n) = sum of (-1)^((u-1)/6) over all solutions of 48*n + 7 = 4*u^2 + 3*v^2 in integers where u == 1 (mod 6) and v == 1 (mod 8).

			1 - x^2 + x^3 - x^4 - 2*x^7 - x^9 + x^10 + x^13 + 2*x^14 + x^15 - x^16 + ...
q^7 - q^103 + q^151 - q^199 - 2*q^343 - q^439 + q^487 + q^631 + 2*q^679 + ...

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. A010815, A143379, A214264.

a[ n_] := SeriesCoefficient[ QPochhammer[ q^2] QPochhammer[ q^8] QPochhammer[ -q^3, q^8] QPochhammer[ -q^5, q^8], {q, 0, n}]

{a(n) = local(s, v); if( n<0, 0, n = 48*n + 7; forstep( u=1, sqrtint( n\4), 2, if( u%3 && issquare( (n - 4*u^2)/3, &v), s += (-1)^((u+1)\6))); s)}

Euler transform of period 16 sequence [ 0, -1, 1, -1, 1, -2, 0, -2, 0, -2, 1, -1, 1, -1, 0, -2, ...].
G.f.: (Sum_{k} (-1)^k * x^(3*k^2 + k)) * (Sum_{k} x^(4*k^2 + k)).
a(n) = A143379(2*n).

A258741 Expansion of f(x^3, x^5) / f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, -1, 1, -1, 2, -2, 2, -3, 4, -5, 5, -6, 8, -9, 10, -12, 15, -17, 19, -22, 26, -30, 33, -38, 45, -51, 56, -64, 74, -83, 92, -104, 119, -133, 147, -165, 187, -208, 229, -256, 288, -319, 351, -390, 435, -481, 528, -584, 649, -715, 783, -863, 954, -1047, 1145

Offset: 0

Michael Somos, Nov 06 2015

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 - x + x^2 - x^3 + 2*x^4 - 2*x^5 + 2*x^6 - 3*x^7 + 4*x^8 - 5*x^9 + ...
G.f. = 1/q - q^15 + q^31 - q^47 + 2*q^63 - 2*q^79 + 2*q^95 - 3*q^111 + ...

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41, 16th equation.

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. A029838, A036016, A082303, A106507, A214264.

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^4, x^4] / (QPochhammer[ -x, x^8] QPochhammer[ -x^7, x^8]), {x, 0, n}];
a[ n_] := SeriesCoefficient[ 1 / Product[ (1 + x^(8 k + 1)) (1 - x^(8 k + 4)) (1 + x^(8 k + 7)), {k, 0, Ceiling[ n/8]}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{1, -1, 0, -1, 0, 0, 1, 0, 1, 0, 0, -1, 0, -1, 1, 0}[[Mod[k, 16, 1]]], {k, n}], {x, 0, n}];

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 0, 1, -1, 0, -1, 0, 0, 1, 0, 1, 0, 0, -1, 0, -1, 1][k%16 + 1]), n))};

Expansion of f(x^4, x^12) / f(x, x^7) where f(, ) is Ramanujan's general theta function.
Euler transform of period 16 sequence [ -1, 1, 0, 1, 0, 0, -1, 0, -1, 0, 0, 1, 0, 1, -1, 0, ...].
G.f.: 1 / (Product_{k>=0} (1 + x^(8*k + 1)) * (1 - x^(8*k + 4)) * (1 + x^(8*k + 7))).
G.f.: (1 + x^4 + x^12 + x^24 + x^40 + ...) / (1 + x + x^7 + x^10 + x^22 + ...). [Ramanujan]
G.f.: 1 - x * (1 - x) / (1 - x^2) + x^4 * (1 - x) * (1 - x^3) / ((1 - x^2) * (1 - x^4)) - ... [Ramanujan]
a(n) = (-1)^n * A036016(n) = A029838(2*n) = A082303(2*n).
Convolution product of A106507 and A214264.

cp's OEIS Frontend

A282893 The difference between the number of partitions of 2n into odd parts (A000009) and the number of partitions of 2n into even parts (A035363).

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A244465 Expansion of f(-x^3, -x^5) in powers of x where f() is Ramanujan's two-variable theta function.

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A246862 Expansion of phi(x) * f(x^3, x^5) in powers of x where phi(), f() are Ramanujan theta functions.

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A214302 Expansion of f(-x^2, -x^4) * f(x^3, x^5) in powers of x where f(,) is Ramanujan's two-variable theta function.

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A258741 Expansion of f(x^3, x^5) / f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function.

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