A229802 -id:A229802 - cp's OEIS frontend

A227216 Expansion of f(-q^2, -q^3)^5 / f(-q)^3 in powers of q where f() is a Ramanujan theta function.

1, 3, 4, 2, 1, 3, 6, 4, 0, -1, 4, 6, 4, 2, 2, 2, 3, 4, 2, 0, 1, 6, 8, 2, 0, 3, 6, 0, -2, 0, 6, 6, 4, 4, 2, 4, 3, 4, 0, -2, 0, 6, 8, 2, 2, -1, 6, 4, 2, 1, 4, 6, 4, 2, 0, 6, 0, 0, 0, 0, 4, 6, 8, 2, 1, 2, 12, 4, -2, -2, 2, 6, 0, 2, 2, 2, 0, 8, 4, 0, 3, 3, 8, 2

Offset: 0

Michael Somos, Sep 21 2013

sign

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

Zagier (2009) refers to Case D corresponding to the Apery numbers (A005258).

			G.f. = 1 + 3*q + 4*q^2 + 2*q^3 + q^4 + 3*q^5 + 6*q^6 + 4*q^7 - q^9 + ...

D. Zagier, Integral solutions of Apery-like recurrence equations, in: Groups and Symmetries: from Neolithic Scots to John McKay, CRM Proc. Lecture Notes 47, Amer. Math. Soc., Providence, RI, 2009, pp. 349-366.

Seiichi Manyama, 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
D. Zagier, Integral solutions of Apery-like recurrence equations.

Cf. A005258, A078905, A229802.

The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692, A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)

A := Basis( ModularForms( Gamma1(5), 1), 20); A[1] + 3*A[2]; /* Michael Somos, Jun 10 2014 */

a[ n_] := If[ n < 1, Boole[ n == 0], Sum[ Re[(3 - I) {1, I, -I, -1, 0}[[ Mod[ d, 5, 1] ]] ], {d, Divisors @ n}]];
a[ n_] := SeriesCoefficient[ QPochhammer[ q]^2 / (QPochhammer[ q, q^5] QPochhammer[ q^4, q^5])^5, {q, 0, n}]; (* Michael Somos, Jun 10 2014 *)

{a(n) = if( n<1, n==0, sumdiv(n, d, real( (3 - I) * [ 0, 1, I, -I, -1][ d%5 + 1])))};

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k)^[ 2, -3, 2, 2, -3][k%5 + 1], 1 + x * O(x^n)), n))};

A = ModularForms( Gamma1(5), 1, prec=20) . basis(); A[0] + 3*A[1]; # Michael Somos, Jun 10 2014

Expansion of f(-q)^2 * (f(-q^5) / f(-q, -q^4))^5 = f(-q^2, -q^3)^2 * (f(-q^5) / f(-q, -q^4))^3 in powers of q where f() is a Ramanujan theta function.
Euler transform of period 5 sequence [ 3, -2, -2, 3, -2, ...].
Moebius transform is period 5 sequence [ 3, 1, -1, -3, 0, ...]. - Michael Somos, Jun 10 2014
G.f. = g(t(q)) where g(), t() are the g.f. for A005258 and A078905.
G.f.: (Product_{k>0} (1 - x^k)^2) / (Product_{k>0} (1 - x^(5*k - 1)) * (1 - x^(5*k - 4)))^5.

A053723 Number of 5-core partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 2, 6, 5, 7, 5, 12, 6, 12, 6, 10, 11, 16, 7, 20, 15, 12, 12, 22, 10, 25, 12, 20, 18, 30, 10, 32, 21, 24, 16, 30, 21, 36, 20, 24, 25, 42, 12, 42, 36, 35, 22, 46, 22, 43, 25, 32, 36, 52, 20, 60, 30, 40, 30, 60, 30, 62, 32, 42, 43, 60, 24, 66, 48, 44, 30, 72, 35, 72

Offset: 0

James Sellers, Feb 11 2000

Number 11 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 2*x^5 + 6*x^6 + 5*x^7 + 7*x^8 + ...
G.f. = q + q^2 + 2*q^3 + 3*q^4 + 5*q^5 + 2*q^6 + 6*q^7 + 5*q^8 + 7*q^9 + ...

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988; see p. 54 (1.52).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Subhajit Bandyopadhyay and Nayandeep Deka Baruah, Arithmetic Identities for Some Analogs of the 5-Core Partition Function, J. Int. Seq. (2024) Vol. 27, Art. No. 24.4.5.
Subhajit Bandyopadhyay and Nayandeep Deka Baruah, Arithmetic Identities for Some Analogs of 5-core Partition Function, arXiv:2409.02034 [math.NT], 2024.
Bruce C. Berndt, The Rogers-Ramanujan continued fraction.
Bruce C. Berndt et al., The Rogers-Ramanujan continued fraction, Journal of Computational and Applied Mathematics, Vol. 105, No. 1-2 (1999), 9-24.
Freeman J. Dyson, Missed opportunities, Bull. Amer. Math. Soc. 78 (1972), 635-652. see pages 636-637.
Frank Garvan, Dongsu Kim, and Dennis Stanton, Cranks and t-cores.
Frank Garvan, Dongsu Kim, and Dennis Stanton, Cranks and t-cores, Inventiones Math. 101 (1990) 1-17.
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers, 2016.

Column t=5 of A175595.
Cf. A053724, A109063, A109064, A138512.
Cf. A227216, A229802, A328717.

a[n_]:=Total[KroneckerSymbol[#, 5]*n/# & /@ Divisors[n]]; Table[a[n], {n, 1, 73}] (* Jean-François Alcover, Jul 26 2011, after PARI prog. *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^5]^5 / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Jul 13 2012 *)
a[ n_] := With[{m = n + 1}, If[ m < 1, 0, DivisorSum[ m, m/# KroneckerSymbol[ 5, #] &]]]; (* Michael Somos, Jul 13 2012 *)

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

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

{a(n) = if( n<0, 0, n++; direuler( p=2, n, 1 / ((1 - p*X) * (1 - kronecker( p, 5) * X)))[n])};

Given g.f. A(x), then B(q) = q * A(q) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = v^3 + 2 * u*v*w + 4 * u*w^2 - u^2*w. - Michael Somos, May 02 2005
G.f.: (1/x) * (Sum_{k>0} Kronecker(k, 5) * x^k / (1 - x^k)^2). - Michael Somos, Sep 02 2005
G.f.: Product_{k>0} (1 - x^(5*k))^5 / (1 - x^k) = 1/x * (Sum_{k>0} k * x^k * (1 - x^k) * (1 - x^(2*k)) / (1 - x^(5*k))). - Michael Somos, Jun 17 2005
G.f.: (1/x) * Sum_{a, b, c, d, e in Z^5} x^((a^2 + b^2 + c^2 + d^2 + e^2) / 10) where a + b + c + d + e = 0, (a, b, c, d, e) == (0, 1, 2, 3, 4) (mod 5). - [Dyson 1972] Michael Somos, Aug 08 2007
Euler transform of period 5 sequence [ 1, 1, 1, 1, -4, ...].
Expansion of q^(-1) * eta(q^5)^5 / eta(q) in powers of q.
a(n) = b(n + 1) where b() is multiplicative with b(5^e) = 5^e, b(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 4 (mod 5), b(p^e) = (p^(e+1) + (-1)^e) / (p + 1) if p == 2, 3 (mod 5).
Convolution inverse of A109063. a(n) = (-1)^n * A138512(n+1).
Convolution of A227216 and A229802. - Michael Somos, Jun 10 2014
G.f. is a period 1 Fourier series which satisfies f(-1 / (5 t)) = (1/5)^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A109064. - Michael Somos, May 17 2015
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A328717. - Amiram Eldar, Nov 23 2023

A239051 Expansion of (f(-q^2, -q^3)^5 - 3 * q * f(-q, -q^4)^5) / f(-q)^3 in powers of q where f() is a Ramanujan theta function.

Original entry on oeis.org

1, 0, 10, -10, 10, 0, 0, 10, 0, -10, 10, 0, 10, -10, 20, -10, 0, 10, -10, 0, 10, 0, 20, -10, 0, 0, 0, 0, 10, 0, 0, 0, 10, -20, 20, 10, 0, 10, 0, -20, 0, 0, 20, -10, 20, -10, 0, 10, -10, 10, 10, 0, 10, -10, 0, 0, 0, 0, 0, 0, 10, 0, 20, -10, 10, -10, 0, 10, 10

Offset: 0

Michael Somos, Jun 13 2014

sign

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

			G.f. = 1 + 10*q^2 - 10*q^3 + 10*q^4 + 10*q^7 - 10*q^9 + 10*q^10 + 10*q^12 + ...

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. A227216, A229802.

Basis( ModularForms( Gamma1(5), 1), 70) [1];

a[ n_] := If[ n < 1, Boole[ n == 0], 10 Sum[ {0, 1, -1, 0, 0}[[ Mod[ d, 5, 1] ]], {d, Divisors @ n}]];

{a(n) = if( n<1, n==0, 10 * sumdiv(n, d, (d%5==2) - (d%5==3)))};

ModularForms( Gamma1(5), 1, prec=70).0;

Moebius transform is period 5 sequence [ 0, 10, -10, 0, 0, ...].
G.f.: 1 + 10 * ( Sum_{k>=0} x^(5*k + 2) / (1 - x^(5*k + 2)) - x^(5*k + 3) / (1 - x^(5*k + 3)) ).
a(n) = A227216(n) - 3 * A229802(n).
a(5*n) = a(n). a(5*n + 1) = 0.

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A227216 Expansion of f(-q^2, -q^3)^5 / f(-q)^3 in powers of q where f() is a Ramanujan theta function.

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A053723 Number of 5-core partitions of n.

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A239051 Expansion of (f(-q^2, -q^3)^5 - 3 * q * f(-q, -q^4)^5) / f(-q)^3 in powers of q where f() is a Ramanujan theta function.

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Comments

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Programs

Magma

Mathematica

PARI

Sage

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