cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A133690 Expansion of (phi(-q) * phi(q^2))^2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -4, 8, -16, 24, -24, 32, -32, 24, -52, 48, -48, 96, -56, 64, -96, 24, -72, 104, -80, 144, -128, 96, -96, 96, -124, 112, -160, 192, -120, 192, -128, 24, -192, 144, -192, 312, -152, 160, -224, 144, -168, 256, -176, 288, -312, 192, -192, 96, -228, 248, -288
Offset: 0

Views

Author

Michael Somos, Sep 20 2007

Keywords

Comments

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

Examples

			G.f. = 1 - 4*q + 8*q^2 - 16*q^3 + 24*q^4 - 24*q^5 + 32*q^6 - 32*q^7 + 24*q^8 - ...

Links

Crossrefs

Cf. A004011, A008438, A097057, A097723, A112610, A133657, A133692.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^2])^2, {q, 0, n}]; (* Michael Somos, Oct 30 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], -4 Which[ OddQ[n], DivisorSigma[ 1, n], Mod[n, 4] > 0, -2 DivisorSigma[1, n/2], True, -6 DivisorSum[n/4, # Mod[#, 2] &]]]; (* Michael Somos, Oct 30 2015 *)
  • PARI
    {a(n) = if( n<1, n==0, -4 * if( n%2, sigma(n), n%4, -2 * sigma(n/2), -6 * sumdiv( n/4, d, (d%2)*d )))};
  • PARI
    {a(n) = local(A); if ( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^2 * eta(x^4 + A)^5 / (eta(x^2 + A)^3 * eta(x^8 + A)^2))^2, n))};

Formula

Expansion of (eta(q)^2 * eta(q^4)^5 / (eta(q^2)^3 * eta(q^8)^2))^2 in powers of q.
Euler transform of period 8 sequence [ -4, 2, -4, -8, -4, 2, -4, -4, ...].
a(n) = -4 * b(n) where b() is multiplicative with b(2) = -2, b(2^e) = -6 if e>1, b(p^e) = (p^(e+1) - 1) / (p - 1) if p>2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 32 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A133657.
G.f.: ( Product_{k>0} (1 - x^k)^2 * (1 + x^(2*k))^3 / (1 + x^(4*k))^2 )^2.
a(n) = (-1)^n * A097057(n). Convolution square of A133692.
a(2*n) = 8 * A046897(n) unless n=0. a(2*n + 1) = A008438(n). a(4*n) = A004011(n). a(4*n + 1) = -4 * A112610(n). a(4*n + 3) = -16 * A097723(n).

A240020 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma(2n-1).

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 5, 3, 5, 6, 6, 7, 7, 8, 8, 8, 9, 9, 10, 10, 11, 5, 5, 11, 12, 12, 13, 5, 13, 14, 6, 6, 14, 15, 15, 16, 16, 17, 7, 7, 17, 18, 12, 18, 19, 19, 20, 8, 8, 20, 21, 21, 22, 22, 23, 32, 23, 24, 24, 25, 7, 25, 26, 10, 10, 26, 27, 27, 28, 8, 8, 28, 29, 11, 11, 29, 30, 30, 31, 31, 32, 12, 26, 12, 32, 33, 9, 9, 33, 34, 34
Offset: 1

Views

Author

Omar E. Pol, Mar 31 2014

Keywords

Comments

Row n lists the parts of the symmetric representation of A008438(n-1).
Also these are the parts from the odd-indexed rows of A237270.
Also these are the parts in the quadrants 1 and 3 of the spiral described in A239660, see example.
Row sums give A008438.
The length of row n is A237271(2n-1).
Both column 1 and the right border are equal to n.
Note that also the sequence can be represented in a quadrant.
We can find the spiral (mentioned above) on the terraces of the stepped pyramid described in A244050. - Omar E. Pol, Dec 07 2016

Examples

			1;
2, 2;
3, 3;
4, 4;
5, 3, 5;
6, 6;
7, 7;
8, 8, 8;
9, 9;
10, 10;
11, 5, 5, 11;
12, 12;
13, 5, 13;
14, 6, 6, 14;
15, 15;
16, 16;
17, 7, 7, 17;
18, 12, 18;
19, 19;
20, 8, 8, 20;
21, 21;
22, 22;
23, 32, 23;
24, 24;
25, 7, 25;
...
Illustration of initial terms (rows 1..8):
.
.                                   _ _ _ _ _ _ _ 7
.                                  |_ _ _ _ _ _ _|
.                                                |
.                                                |_ _
.                                   _ _ _ _ _ 5      |_
.                                  |_ _ _ _ _|         |
.                                            |_ _ 3    |_ _ _ 7
.                                            |_  |         | |
.                                   _ _ _ 3    |_|_ _ 5    | |
.                                  |_ _ _|         | |     | |
.                                        |_ _ 3    | |     | |
.                                          | |     | |     | |
.                                   _ 1    | |     | |     | |
.     _       _       _       _    |_|     |_|     |_|     |_|
.    | |     | |     | |     | |
.    | |     | |     | |     |_|_ _
.    | |     | |     | |    2  |_ _|
.    | |     | |     |_|_     2
.    | |     | |    4    |_
.    | |     |_|_ _        |_ _ _ _
.    | |    6      |_      |_ _ _ _|
.    |_|_ _ _        |_   4
.   8      | |_ _      |
.          |_    |     |_ _ _ _ _ _
.            |_  |_    |_ _ _ _ _ _|
.           8  |_ _|  6
.                  |
.                  |_ _ _ _ _ _ _ _
.                  |_ _ _ _ _ _ _ _|
.                 8
.
The figure shows the quadrants 1 and 3 of the spiral described in A239660.
For n = 5 we have that 2*5 - 1 = 9 and the 9th row of A237593 is [5, 2, 2, 2, 2, 5] and the 8th row of A237593 is [5, 2, 1, 1, 2, 5] therefore between both symmetric Dyck paths there are three regions (or parts) of sizes [5, 3, 5], so row 5 is [5, 3, 5], see the third arm of the spiral in the first quadrant.
The sum of divisors of 9 is 1 + 3 + 9 = A000203(9) = 13. On the other hand the sum of the parts of the symmetric representation of sigma(9) is 5 + 3 + 5 = 13, equaling the sum of divisors of 9.

Crossrefs

Cf. A000203, A005408, A008438, A112610, A196020, A236104, A237048, A237270, A237271, A237591, A237593, A239053, A239660, A239931, A239933, A244050, A245092, A262626.

A257214 E.g.f.: C(x) = Sum_{n>=0} cosh((2*n+1)*x) * x^n / (1 + x^(2*n+1)).

Original entry on oeis.org

1, 0, 5, 24, 337, 3280, 46501, 811496, 15270977, 318449952, 7554700261, 194401167928, 5484157128913, 167431552506608, 5496127228989125, 193614639911456520, 7265814918674507521, 289758831638674507840, 12237733598089127162437, 545392221565792906192472, 25589486575413268343127761, 1260584085915542118144276240
Offset: 0

Views

Author

Paul D. Hanna, Apr 18 2015

Keywords

Examples

			E.g.f.: C(x) = 1 + 5*x^2/2! + 24*x^3/3! + 337*x^4/4! + 3280*x^5/5! +...
where
C(x) = cosh(x)/(1+x) + cosh(3*x)*x/(1+x^3) + cosh(5*x)*x^2/(1+x^5) + cosh(7*x)*x^3/(1+x^7) + cosh(9*x)*x^4/(1+x^9) + cosh(11*x)*x^5/(1+x^11) +...
RELATED SERIES.
The dual Lambert series
S(x) = sinh(x)/(1-x) + sinh(3*x)*x/(1-x^3) + sinh(5*x)*x^2/(1-x^5) + sinh(7*x)*x^3/(1-x^7) + sinh(9*x)*x^4/(1-x^9) + sinh(11*x)*x^5/(1-x^11) +...
S(x) = x + 8*x^2/2! + 37*x^3/3! + 304*x^4/4! + 4081*x^5/5! +...
is related by
C(x)^2 - S(x)^2 = R(x)^2 = 1 + 4*x^2 + 6*x^4 + 8*x^6 + 13*x^8 + 12*x^10 + 14*x^12 + 24*x^14 + 18*x^16 + 20*x^18 + 32*x^20 +...+ A008438(n)*x^(2*n) +...
such that
R(x)^(1/2) = 1 + x^2 + x^6 + x^12 + x^20 + x^30 + x^42 +...+ x^(n^2+n) +...
The squares of these related series begin:
C(x)^2 = 1 + 10*x^2/2! + 48*x^3/3! + 824*x^4/4! + 8960*x^5/5! + 155072*x^6/6! + 2877952*x^7/7! + 60328704*x^8/8! + 1395081216*x^9/9! +...
S(x)^2 = 2*x^2/2! + 48*x^3/3! + 680*x^4/4! + 8960*x^5/5! + 149312*x^6/6! + 2877952*x^7/7! + 59804544*x^8/8! + 1395081216*x^9/9! +...
R(x)^2 = C(x)^2 - S(x)^2 = 1 + 4*x^2 + 6*x^4 + 8*x^6 + 13*x^8 + 12*x^10 + 14*x^12 + 24*x^14 + 18*x^16 + 20*x^18 + 32*x^20 +...
The normalized series begin
C(x)/R(x) = 1 + x^2/2! + 24*x^3/3! + 289*x^4/4! + 2320*x^5/5! + 27361*x^6/6! + 596456*x^7/7! + 11600065*x^8/8! +...
S(x)/R(x) = x + 8*x^2/2! + 25*x^3/3! + 112*x^4/4! + 2961*x^5/5! + 41784*x^6/6! + 557929*x^7/7! + 10393184*x^8/8! +...
(C(x) + S(x))/R(x) = 1 + x + 9*x^2/2! + 49*x^3/3! + 401*x^4/4! + 5281*x^5/5! + 69145*x^6/6! + 1154385*x^7/7! + 21993249*x^8/8! +...
where
C(x) + S(x) = 1 + x + 13*x^2/2! + 61*x^3/3! + 641*x^4/4! + 7361*x^5/5! + 97885*x^6/6! + 1649229*x^7/7! + 30854689*x^8/8! +...
C(x) + S(x) = Sum_{n>=0} [exp((2*n+1)*x)*x^n/(1-x^(4*n+2)) - exp(-(2*n+1)*x)*x^(3*n+1)/(1-x^(4*n+2))].

Links

Crossrefs

Cf. A257215, A008438.

Programs

  • PARI
    {a(n)=local(A = sum(m=0,n, cosh((2*m+1)*x +x*O(x^n)) * x^m/(1+x^(2*m+1)) )); n!*polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

E.g.f. C(x) satisfies:
(1) C(x)^2 - S(x)^2 = R(x)^2,
(2) C(x) * (C(x)/R(x))' = S(x) * (S(x)/R(x))',
where
(a) R(x) = [ Sum_{n>=0} x^(n*(n+1)) ]^2, and
(b) S(x) = Sum_{n>=0} sinh((2*n+1)*x) * x^n / (1 - x^(2*n+1)) the e.g.f. of A257215.

A257215 E.g.f.: S(x) = Sum_{n>=0} sinh((2*n+1)*x) * x^n / (1 - x^(2*n+1)).

Original entry on oeis.org

1, 8, 37, 304, 4081, 51384, 837733, 15583712, 324393985, 7669671400, 195589720261, 5509114219536, 168051665376817, 5506719600441752, 193872344999763781, 7271477485665147328, 289936454250117720193, 12242148798010459653576, 545520427163375125201381, 25593712286164576808576240
Offset: 1

Views

Author

Paul D. Hanna, Apr 18 2015

Keywords

Examples

			E.g.f.: S(x) = x + 8*x^2/2! + 37*x^3/3! + 304*x^4/4! + 4081*x^5/5! +...
where
S(x) = sinh(x)/(1-x) + sinh(3*x)*x/(1-x^3) + sinh(5*x)*x^2/(1-x^5) + sinh(7*x)*x^3/(1-x^7) + sinh(9*x)*x^4/(1-x^9) + sinh(11*x)*x^5/(1-x^11) +...
RELATED SERIES.
The dual Lambert series
C(x) = cosh(x)/(1+x) + cosh(3*x)*x/(1+x^3) + cosh(5*x)*x^2/(1+x^5) + cosh(7*x)*x^3/(1+x^7) + cosh(9*x)*x^4/(1+x^9) + cosh(11*x)*x^5/(1+x^11) +...
C(x) = 1 + 5*x^2/2! + 24*x^3/3! + 337*x^4/4! + 3280*x^5/5! +...
is related by
C(x)^2 - S(x)^2 = R(x)^2 = 1 + 4*x^2 + 6*x^4 + 8*x^6 + 13*x^8 + 12*x^10 + 14*x^12 + 24*x^14 + 18*x^16 + 20*x^18 + 32*x^20 +... + A008438(n)*x^(2*n) + ...
such that
R(x)^(1/2) = 1 + x^2 + x^6 + x^12 + x^20 + x^30 + x^42 +...+ x^(n^2+n) +...
The squares of these related series begin:
C(x)^2 = 1 + 10*x^2/2! + 48*x^3/3! + 824*x^4/4! + 8960*x^5/5! + 155072*x^6/6! + 2877952*x^7/7! + 60328704*x^8/8! + 1395081216*x^9/9! +...
S(x)^2 = 2*x^2/2! + 48*x^3/3! + 680*x^4/4! + 8960*x^5/5! + 149312*x^6/6! + 2877952*x^7/7! + 59804544*x^8/8! + 1395081216*x^9/9! +...
R(x)^2 = C(x)^2 - S(x)^2 = 1 + 4*x^2 + 6*x^4 + 8*x^6 + 13*x^8 + 12*x^10 + 14*x^12 + 24*x^14 + 18*x^16 + 20*x^18 + 32*x^20 +...
The normalized series begin
C(x)/R(x) = 1 + x^2/2! + 24*x^3/3! + 289*x^4/4! + 2320*x^5/5! + 27361*x^6/6! + 596456*x^7/7! + 11600065*x^8/8! +...
S(x)/R(x) = x + 8*x^2/2! + 25*x^3/3! + 112*x^4/4! + 2961*x^5/5! + 41784*x^6/6! + 557929*x^7/7! + 10393184*x^8/8! +...
(C(x) + S(x))/R(x) = 1 + x + 9*x^2/2! + 49*x^3/3! + 401*x^4/4! + 5281*x^5/5! + 69145*x^6/6! + 1154385*x^7/7! + 21993249*x^8/8! +...
where
C(x) + S(x) = 1 + x + 13*x^2/2! + 61*x^3/3! + 641*x^4/4! + 7361*x^5/5! + 97885*x^6/6! + 1649229*x^7/7! + 30854689*x^8/8! +...
C(x) + S(x) = Sum_{n>=0} [exp((2*n+1)*x)*x^n/(1-x^(4*n+2)) - exp(-(2*n+1)*x)*x^(3*n+1)/(1-x^(4*n+2))].

Links

Crossrefs

Cf. A257214, A008438.

Programs

  • PARI
    {a(n)=local(A = sum(m=0,n, sinh((2*m+1)*x +x*O(x^n)) * x^m/(1-x^(2*m+1)) )); n!*polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))

Formula

E.g.f. S(x) satisfies:
(1) C(x)^2 - S(x)^2 = R(x)^2,
(2) S(x) * (S(x)/R(x))' = C(x) * (C(x)/R(x))',
where
(a) R(x) = [ Sum_{n>=0} x^(n*(n+1)) ]^2, and
(b) C(x) = Sum_{n>=0} cosh((2*n+1)*x) * x^n / (1 + x^(2*n+1)) the e.g.f of A257214.

A346869 Sum of all divisors, except the smallest and the largest of every number, of the first n odd numbers.

Original entry on oeis.org

0, 0, 0, 0, 3, 3, 3, 11, 11, 11, 21, 21, 26, 38, 38, 38, 52, 64, 64, 80, 80, 80, 112, 112, 119, 139, 139, 155, 177, 177, 177, 217, 235, 235, 261, 261, 261, 309, 327, 327, 366, 366, 388, 420, 420, 440, 474, 498, 498, 554, 554, 554, 640, 640, 640, 680, 680, 708, 772, 796
Offset: 1

Views

Author

Omar E. Pol, Aug 18 2021

Keywords

Comments

Partial sums of the odd-indexed terms of Chowla's function A048050.
a(n) has a symmetric representation.

Crossrefs

Partial sums of A346879.
Cf. A000203, A005408, A008438, A048050, A237593, A245092, A244049, A326123, A346870.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=1, 0,
      a(n-1)+numtheory[sigma](2*n-1)-2*n)
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Aug 19 2021
  • Mathematica
    s[1] = 0; s[n_] := DivisorSigma[1, 2*n - 1] - 2*n; Accumulate @ Array[s, 50] (* Amiram Eldar, Aug 19 2021 *)
Accumulate[Join[{0},Table[DivisorSigma[1,n]-n-1,{n,3,151,2}]]] (* Harvey P. Dale, Jul 29 2023 *)
  • Python
    from sympy import divisors
from itertools import accumulate
def A346879(n): return sum(divisors(2*n-1)[1:-1])
def aupton(nn): return list(accumulate(A346879(n) for n in range(1, nn+1)))
print(aupton(60)) # Michael S. Branicky, Aug 19 2021

A346879 Sum of the divisors, except the smallest and the largest, of the n-th odd number.

Original entry on oeis.org

0, 0, 0, 0, 3, 0, 0, 8, 0, 0, 10, 0, 5, 12, 0, 0, 14, 12, 0, 16, 0, 0, 32, 0, 7, 20, 0, 16, 22, 0, 0, 40, 18, 0, 26, 0, 0, 48, 18, 0, 39, 0, 22, 32, 0, 20, 34, 24, 0, 56, 0, 0, 86, 0, 0, 40, 0, 28, 64, 24, 11, 44, 30, 0, 46, 0, 26, 104, 0, 0, 50, 24, 34, 80, 0, 0, 80, 36
Offset: 1

Views

Author

Omar E. Pol, Aug 18 2021

Keywords

Comments

a(n) has a symmetric representation.

Examples

			For n = 5 the 5th odd number is 9 and the divisors of 9 are [1, 3, 9] and the sum of the divisors of 9 except the smaller and the largest is 3, so a(5) = 3.
For n = 6 the 6th odd number is 11 and the divisors of 11 are [1, 11] and the sum of the divisors of 11 except the smaller and the largest is 0, so a(6) = 0.

Crossrefs

Bisection of A048050.
Partial sums give A346869.
Cf. A000203, A005408, A008438, A237593, A245092, A244049, A326123, A346880.

Programs

  • Mathematica
    a[1] = 0; a[n_] := DivisorSigma[1, 2*n - 1] - 2*n; Array[a, 100] (* Amiram Eldar, Aug 19 2021 *)
  • Python
    from sympy import divisors
def a(n): return sum(divisors(2*n-1)[1:-1])
print([a(n) for n in range(1, 79)]) # Michael S. Branicky, Aug 19 2021

Formula

a(n) = A048050(2*n-1).

A364092 Sum of divisors of 5*n-1 of form 5*k+1.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 1, 12, 1, 7, 1, 17, 1, 1, 1, 28, 1, 1, 12, 27, 1, 7, 1, 32, 1, 1, 1, 59, 1, 12, 1, 42, 1, 7, 1, 47, 22, 1, 1, 58, 12, 1, 1, 73, 1, 33, 1, 62, 1, 1, 1, 84, 1, 1, 32, 72, 1, 28, 1, 93, 1, 1, 12, 124, 1, 1, 1, 87, 1, 7, 1, 118, 42, 12, 1, 119, 1, 1, 22, 102, 1, 53, 1, 107, 12, 32, 1
Offset: 1

Views

Author

Seiichi Manyama, Jul 04 2023

Keywords

Crossrefs

Cf. A364093, A364094, A364095.
Cf. A008438, A363514.
Cf. A284097, A359233, A364096.

Programs

  • Mathematica
    a[n_] := DivisorSum[5*n - 1, # &, Mod[#, 5] == 1 &]; Array[a, 100] (* Amiram Eldar, Jul 17 2023 *)
  • PARI
    a(n) = sumdiv(5*n-1, d, (d%5==1)*d);

Formula

a(n) = A284097(5*n-1).
G.f.: Sum_{k>0} (5*k-4) * x^(4*k-3) / (1 - x^(5*k-4)).

A225699 Numerators of coefficients arising from q-expansion of Integrate[eta[q^4]^8/eta[q^2]^4, q]/q where eta is the Dedekind eta function.

Original entry on oeis.org

1, 1, 1, 1, 13, 1, 1, 3, 1, 1, 16, 1, 31, 10, 1, 1, 24, 4, 1, 7, 1, 1, 39, 1, 57, 18, 1, 9, 40, 1, 1, 13, 14, 1, 48, 1, 1, 31, 16, 1, 121, 1, 54, 15, 1, 28, 64, 5, 1, 39, 1, 1, 96
Offset: 0

Views

Author

N. J. A. Sloane, Jun 01 2013

Keywords

Comments

Gosper observes that A225699/A225700 = A008438/(2,4,6,8,10,...) and hence the coefficient of q^k in the q-expansion is 1 iff k is an odd prime (see Example section below).
Note that, as usual in the OEIS, the q-expansion has been normalized here to avoid having every other term be zero.

Examples

			q/2 + q^3 + q^5 + q^7 + (13*q^9)/10 + q^11 + q^13 + (3*q^15)/2 + q^17 + q^19 + (16*q^21)/11 + q^23 + (31*q^25)/26 + (10*q^27)/7 + q^29 + q^31 + (24*q^33)/17 + (4*q^35)/3 + q^37 + (7*q^39)/5 + q^41 + q^43 + (39*q^45)/23 + q^47 + (57*q^49)/50 + (18*q^51)/13 + q^53 + (9*q^55)/7 + (40*q^57)/29 + q^59 + q^61 + (13*q^63)/8 + (14*q^65)/11 + q^67 + (48*q^69)/35 + q^71 + q^73 + (31*q^75)/19 + (16*q^77)/13 + q^79 + (121*q^81)/82 + q^83 + (54*q^85)/43 + (15*q^87)/11 + q^89 + (28*q^91)/23 + (64*q^93)/47 + (5*q^95)/4 + q^97 + (39*q^99)/25 + q^101 + q^103 + (96*q^105)/53 + ...

References

  • R. W. Gosper, Posting to the Math Fun Mailing List, Jun 01 2013

Crossrefs

Cf. A225700. See A008438 for eta[q^4]^8/eta[q^2]^4.

A225700 Denominators of coefficients arising from q-expansion of Integrate[eta[q^4]^8/eta[q^2]^4, q]/q where eta is the Dedekind eta function.

Original entry on oeis.org

2, 1, 1, 1, 10, 1, 1, 2, 1, 1, 11, 1, 26, 7, 1, 1, 17, 3, 1, 5, 1, 1, 23, 1, 50, 13, 1, 7, 29, 1, 1, 8, 11, 1, 35, 1, 1, 19, 13, 1, 82, 1, 43, 11, 1, 23, 47, 4, 1, 25, 1, 1, 53
Offset: 0

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Author

N. J. A. Sloane, Jun 01 2013

Keywords

Comments

Gosper observes that A225699/A225700 = A008438/(2,4,6,8,10,...) and hence the coefficient of q^k in the q-expansion is 1 iff k is an odd prime (see Example section below).
Note that, as usual in the OEIS, the q-expansion has been normalized here to avoid having every other term be zero.

Examples

			q/2 + q^3 + q^5 + q^7 + (13*q^9)/10 + q^11 + q^13 + (3*q^15)/2 + q^17 + q^19 + (16*q^21)/11 + q^23 + (31*q^25)/26 + (10*q^27)/7 + q^29 + q^31 + (24*q^33)/17 + (4*q^35)/3 + q^37 + (7*q^39)/5 + q^41 + q^43 + (39*q^45)/23 + q^47 + (57*q^49)/50 + (18*q^51)/13 + q^53 + (9*q^55)/7 + (40*q^57)/29 + q^59 + q^61 + (13*q^63)/8 + (14*q^65)/11 + q^67 + (48*q^69)/35 + q^71 + q^73 + (31*q^75)/19 + (16*q^77)/13 + q^79 + (121*q^81)/82 + q^83 + (54*q^85)/43 + (15*q^87)/11 + q^89 + (28*q^91)/23 + (64*q^93)/47 + (5*q^95)/4 + q^97 + (39*q^99)/25 + q^101 + q^103 + (96*q^105)/53 + ...

References

  • R. W. Gosper, Posting to the Math Fun Mailing List, Jun 01 2013

Crossrefs

Cf. A225700. See A008438 for eta[q^4]^8/eta[q^2]^4.

A228746 Expansion of 8 * phi(q)^4 - 7 * phi(-q)^4 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 120, 24, 480, 24, 720, 96, 960, 24, 1560, 144, 1440, 96, 1680, 192, 2880, 24, 2160, 312, 2400, 144, 3840, 288, 2880, 96, 3720, 336, 4800, 192, 3600, 576, 3840, 24, 5760, 432, 5760, 312, 4560, 480, 6720, 144, 5040, 768, 5280, 288, 9360, 576, 5760, 96, 6840
Offset: 0

Views

Author

Michael Somos, Sep 02 2013

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution with A005875 is A004008.

Examples

			G.f. = 1 + 120*q + 24*q^2 + 480*q^3 + 24*q^4 + 720*q^5 + 96*q^6 + 960*q^7 + ...

Links

Crossrefs

Cf. A004008, A004011, A005875, A008438, A212002, A228745.
Cf. A000122, A000700, A010054, A121373.

Programs

  • Magma
    A := Basis( ModularForms( Gamma0(4), 2), 50); A[1] + 120*A[2]; /* Michael Somos, Aug 21 2014 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ (8 EllipticTheta[ 3, 0, q]^4 - 7 EllipticTheta[ 4, 0, q]^4), {q, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = sum(k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n)); polcoeff( 8 * A^4 - 7 * subst(A, x, -x)^4, n))};

Formula

a(n) = 120 * b(n) with b() multiplicative where b(2^e) = 1/5 if e>1, b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)), if p>2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 32 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A228745.
G.f.: 8 * (Sum_{k in Z} x^k^2)^4 - 7 * (Sum_{k in Z} (-x)^k^2)^4 .
a(2*n) = A004011(n). a(2*n + 1) = 120 * A008438(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 4*Pi^2 = 39.478417... (A212002). - Amiram Eldar, Dec 29 2023
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