A187154 -id:A187154 - cp's OEIS frontend

A115671 Number of partitions of n into parts not congruent to 0, 2, 12, 14, 16, 18, 20, 30 (mod 32).

1, 1, 1, 2, 3, 4, 6, 8, 11, 15, 20, 26, 34, 44, 56, 72, 91, 114, 143, 178, 220, 272, 334, 408, 498, 605, 732, 884, 1064, 1276, 1528, 1824, 2171, 2580, 3058, 3616, 4269, 5028, 5910, 6936, 8124, 9498, 11088, 12922, 15034, 17468, 20264, 23472, 27154, 31369

Offset: 0

Michael Somos, Jan 29 2006

nonn

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

Andrews (1987) refers to this sequence as p(S, n) where S is the set in equation (1) on page 437.

			G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 11*x^8 + 15*x^9 + ...
a(5) = 4 since 5 = 4 + 1 = 3 + 1 + 1 = 1 + 1 + 1 + 1 + 1 in 4 ways.
a(6) = 6 since 6 = 5 + 1 = 4 + 1 + 1 = 3 + 3 = 3 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1 + 1 in 6 ways.

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..120 from Reinhard Zumkeller)
G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 32. MR0858826 (88b:11063).
G. E. Andrews, Unsolved Problems: Further Problems on Partitions, Amer. Math. Monthly 94 (1987), no. 5, 437-439.
Mircea Merca, The bisectional pentagonal number theorem, Journal of Number Theory, Volume 157, December 2015, Pages 223-232.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A080054, A187154, A208851, A208856, A218171.

a115671 = p [x | x <- [0..], (mod x 32) `notElem` [0,2,12,14,16,18,20,30]]
   where p _          0 = 1
         p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Mar 03 2012

a[ n_] := SeriesCoefficient[ (QPochhammer[ -q] / QPochhammer[ q] + 1) / 2, {q, 0, n}]; (* Michael Somos, Nov 09 2014 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2]^3 / QPochhammer[ q]^2 / QPochhammer[ q^4] + 1) / 2, {q, 0, n}]; (* Michael Somos, Nov 09 2014 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 + eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A))) / 2, n))};

Expansion of (f(q) / f(-q) + 1) / 2 in powers of q where f() is a Ramanujan theta function.
Expansion of f(q^6, q^10) / f(-q, -q^3) = (f(q^22, q^26) - q^2 * f(q^10, q^38)) / f(-q, -q^2) in powers of x where f() is Ramanujan's two-variable theta function.
Euler transform of period 32 sequence [ 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, ...].
Given g.f. A(x), then B(x) = (2*A(x) - 1)^2 = g.f. A007096 satisfies 0 = f(B(x), B(x^2)) where f(u, v) = 1 + u^2 - 2 * u * v^2.
G.f. (1 + sqrt( theta_3(x) / theta_4(x))) / 2 = (Sum_{k} x^(8*k^2 - 2*k)) / (Sum_{k} (-x)^(2*k^2 - k)) = (Sum_{k} x^(24*n^2 + 2*n) - x^(24*n^2 + 14*n + 2)) / (Product_{k>0} 1 - x^k).
2 * a(n) = A080054(n) unless n = 0. a(2*n + 2) = A208851(n). a(2*n + 1) = A187154(n). a(n + 1) = A208856(n).

A210063 Expansion of psi(x^4) / phi(x) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -2, 4, -8, 15, -26, 44, -72, 114, -178, 272, -408, 605, -884, 1276, -1824, 2580, -3616, 5028, -6936, 9498, -12922, 17468, -23472, 31369, -41700, 55156, -72616, 95172, -124202, 161436, -209016, 269616, -346562, 443952, -566856, 721530, -915642, 1158608

Offset: 0

Michael Somos, Mar 16 2012

sign

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

			1 - 2*x + 4*x^2 - 8*x^3 + 15*x^4 - 26*x^5 + 44*x^6 - 72*x^7 + 114*x^8 + ...
q - 2*q^3 + 4*q^5 - 8*q^7 + 15*q^9 - 26*q^11 + 44*q^13 - 72*q^15 + 114*q^17 + ...

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. A187154, A208589, A210030.

CoefficientList[Series[x^(-1/2) * EllipticTheta[2, 0, x^2] / (2*EllipticTheta[3, 0, x]), {x, 0, 50}], x] (* Vaclav Kotesovec, Nov 17 2017 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A) * eta(x^8 + A)^2 / eta(x^2 + A)^5, n))}

Expansion of q^(-1/2) * eta(q)^2 * eta(q^4) * eta(q^8)^2 / eta(q^2)^5 in powers of q.
Euler transform of period 8 sequence [ -2, 3, -2, 2, -2, 3, -2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 8^(-1/2) * g(t) where q = exp(2 Pi i t) and g() is g.f. for A210030.
a(n) = (-1)^n * A187154(n). Convolution inverse of A208589.
a(n) ~ (-1)^n * exp(sqrt(n)*Pi) / (16*n^(3/4)). - Vaclav Kotesovec, Nov 17 2017

A208856 Partitions of n into parts not congruent to 0, +-4, +-6, +-10, 16 (mod 32).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 11, 15, 20, 26, 34, 44, 56, 72, 91, 114, 143, 178, 220, 272, 334, 408, 498, 605, 732, 884, 1064, 1276, 1528, 1824, 2171, 2580, 3058, 3616, 4269, 5028, 5910, 6936, 8124, 9498, 11088, 12922, 15034, 17468, 20264, 23472, 27154, 31369, 36189

Offset: 0

Michael Somos, Mar 02 2012

nonn

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

Andrews (1987) refers to this sequence as p(T, n) where T is the set in equation (1) on page 437.

			1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 8*x^6 + 11*x^7 + 15*x^8 + 20*x^9 + ...
a(5) = 6 since  5 = 3 + 2 = 3 + 1 + 1 = 2 + 2 + 1 = 2 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1 in 6 ways.
a(6) = 8 since  5 + 1 = 3 + 3 = 3 + 2 + 1 = 3 + 1 + 1 + 1 = 2 + 2 + 2 = 2 + 2 + 1 + 1 = 2 + 1 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1 + 1 in 8 ways.

G. C. Greubel, Table of n, a(n) for n = 0..1000
G. E. Andrews, Unsolved Problems: Further Problems on Partitions, Amer. Math. Monthly 94 (1987), no. 5, 437-439.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A080054, A115671, A187154, A208851.

A208856[n_] := SeriesCoefficient[(1/(2*q))*((QPochhammer[-q, -q]/ QPochhammer[q, q]) - 1), {q, 0, n}]; Table[A208856[n], {n,0,50}] (* G. C. Greubel, Jun 19 2017 *)

{a(n) = local(A); if( n<0, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)) - 1) / 2, n))}

Expansion of (f(x) / f(-x) - 1) / (2 * x) in powers of x where f() is a Ramanujan theta function.
Expansion of (f(x^14, x^34) - x^4 * f(x^2, x^46)) / f(-x, -x^2) in powers of x where f() is Ramanujan's two-variable theta function.
Euler transform of period 32 sequence [ 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, ...].
a(n) = A115671(n + 1). 2 * a(n) = A080054(n + 1). a(2*n) = A187154(n). a(2*n + 1) = A208851(n).

A093085 Expansion of phi(-x) / psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -2, 0, 0, 1, 2, 0, 0, -1, -4, 0, 0, 0, 6, 0, 0, 1, -8, 0, 0, 0, 12, 0, 0, -1, -18, 0, 0, -1, 24, 0, 0, 2, -32, 0, 0, 1, 44, 0, 0, -2, -58, 0, 0, -1, 76, 0, 0, 2, -100, 0, 0, 1, 128, 0, 0, -3, -164, 0, 0, -1, 210, 0, 0, 4, -264, 0, 0, 2, 332, 0, 0, -5, -416, 0, 0, -2, 516, 0, 0, 5, -640, 0, 0, 2, 790, 0, 0, -6, -968

Offset: 0

Michael Somos, Mar 20 2004, Oct 22 2007

sign

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

eta(q^2) * eta(q^8)^6 = eta(q)^2 * eta(q^4)^2 * eta(q^8) * eta(q^16)^2 + 2 * eta(q^2) * eta(q^4)^2 * eta(q^16)^4 is equivalent to the a(4*n), ..., a(4*n + 3) results.

			G.f. = 1 - 2*x + x^4 + 2*x^5 - x^8 - 4*x^9 + 6*x^13 + x^16 - 8*x^17 + 12*x^21 - ...
G.f. = 1/q - 2*q + q^7 + 2*q^9 - q^15 - 4*q^17 + 6*q^25 + q^31 - 8*q^33 + ...

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, A029839, A083365, A131124, A187154.

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

{a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k)^[ 0, 2, 1, 2, 2, 2, 1, 2][1 + k%8], 1 + x * O(x^n)), n))}

{a(n) = my(A, A2, m); if( n<0, 0, A = x + O(x^2); m=1; while( m<=n, m*=2; A = subst(A, x, x^2); A = 4*A + 16*A^2 + (1 + 8*A) * sqrt(A + 4*A^2)); polcoeff( sqrt(x / A), n))}

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A) / (eta(x^2 + A) * eta(x^8 + A)^2), n))}

Expansion of q^(1/2) * eta(q)^2 * eta(q^4) / (eta(q^2) * eta(q^8)^2) in powers of q.
Euler transform of period 8 sequence [ -2, -1, -2, -2, -2, -1, -2, 0, ...].
Given g.f. A(x), then B(q) = A(q)^2 / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u * (u + 8) * (v + 4) - v^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 32^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A187154.
G.f.: Product_{k>0} (1 - x^k)^2 / ((1 - x^(4*k - 2)) * (1 - x^(8*k))^2).
a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A029838(n). a(4*n + 1) =  -2 * A083365(n). Convolution square is A131124. Convolution inverse is A187154.

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A115671 Number of partitions of n into parts not congruent to 0, 2, 12, 14, 16, 18, 20, 30 (mod 32).

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A210063 Expansion of psi(x^4) / phi(x) in powers of x where phi(), psi() are Ramanujan theta functions.

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A208856 Partitions of n into parts not congruent to 0, +-4, +-6, +-10, 16 (mod 32).

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A093085 Expansion of phi(-x) / psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.

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