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.

Showing 1-6 of 6 results.

A121373 Expansion of f(x) = f(x, -x^2) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 1, -1, 0, 0, -1, 0, -1, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 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, -1, 0, 0, 0, 0, 0, 0, 1, 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
Offset: 0

Views

Author

Michael Somos, Jul 24 2006

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
This is an example of the quintuple product identity in the form f(a*b^4, a^2/b) - (a/b) * f(a^4*b, b^2/a) = f(-a*b, -a^2*b^2) * f(-a/b, -b^2) / f(a, b) where a = -x^3, b = -x. - Michael Somos, Jul 11 2012
Number 5 of the 14 primitive eta-products which are holomorphic modular forms of weight 1/2 listed by D. Zagier on page 30 of "The 1-2-3 of Modular Forms". - Michael Somos, May 04 2016

Examples

			G.f. = 1 + x - x^2 - x^5 - x^7 - x^12 + x^15 + x^22 + x^26 + x^35 + ...
G.f. = q + q^25 - q^49 - q^121 - q^169 - q^289 + q^361 + q^529 + ...

Links

Crossrefs

Cf. A010815, A080995, A247133, A247223.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ Product[ 1 - (-x)^k, {k, n}], {x, 0, n}]; (* Michael Somos, Nov 14 2011 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x], {x, 0, n}]; (* Michael Somos, Jul 06 2013 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 1, Pi/12, x^4] + EllipticTheta[ 2, Pi/12, x^4]) / Sqrt[6], {x, 0, 24 n + 1}] // Simplify; (* Michael Somos, Mar 20 2015 *)
  • PARI
    {a(n) = if( issquare( 24*n + 1, &n), kronecker( 6, n))};
  • PARI
    {a(n) = if( n<0, 0, polcoeff( eta( -x + x * O(x^n)), n))};

Formula

Expansion of q^(-1/4) * (theta_1( Pi/12, q) + theta_2( Pi/12, q)) / sqrt(6) in powers of q^6. - Michael Somos, Jul 06 2013
Expansion of q^(-1/24) * eta(q^2)^3 / (eta(q) * eta(q^4)) in powers of q.
Euler transform of period 4 sequence [1, -2, 1, -1, ...].
a(n) = b(24*n + 1) where b() is multiplicative with b(p^2e) = (-1)^e if p == 7, 11, 13, 17 (mod 24), b(p^2e) = +1 if p == 1, 5, 19, 23 (mod 24) and b(p^(2e-1)) = b(2^e) = b(3^e) = 0 if e>0.
G.f.: (1 + x) * (1 - x^2) * (1 + x^3) * (1 - x^4) * ...
G.f.: 1 + x - x^2*(1 + x) + x^3*(1 + x)*(1 - x^2) - x^4*(1 + x)*(1 - x^2)*(1 + x^3) + ...
a(5*n + 3) = a(5*n + 4) = 0. a(25*n + 1) = a(n).
G.f.: Sum_{k>=0} a(k) * x^(24*k + 1) = Sum_{k in Z} (-1)^floor((k+1)/2) * x^(6*k + 1)^2.
a(n) = (-1)^n * A010815(n). |a(n)| = A080995(n).
Expansion of f(-x^5, -x^7) + x * f(-x, -x^11) in powers of x. - Michael Somos, Jan 10 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = 48^(1/2) (t/i)^(1/2) f(t) where q = exp(2 Pi i t). - Michael Somos, May 05 2016
G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - (-x)^k))). - Ilya Gutkovskiy, Jun 08 2018

A210964 Column 10 of square array A195825. Also column 1 of triangle A210954. Also 1 together with the row sums of triangle A210954.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 7, 10, 12, 13, 13, 13, 13, 13, 13, 13, 14, 16, 21, 27, 32, 34, 35, 35, 35, 35, 35, 36, 38, 44, 54, 67, 77, 83, 85, 86, 86, 86, 87, 89, 95, 107, 128, 152, 173, 185, 191, 193, 194, 195
Offset: 0

Views

Author

Omar E. Pol, Jun 16 2012

Keywords

Comments

Note that this sequence contains five plateaus: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [4, 4, 4, 4, 4, 4, 4, 4, 4], [13, 13, 13, 13, 13, 13, 13], [35, 35, 35, 35, 35], [86, 86, 86]. For more information see A210843 and other sequences of this family. - Omar E. Pol, Jun 29 2012

Links

Crossrefs

Cf. A000041, A006950, A036820, A057077, A195818, A195825, A195848, A195849, A195850, A195851, A195852, A196933, A210944, A210954, A211970, A211971.
Cf. A247133.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[1 / ((1 - x^(12*k)) * (1 - x^(12*k-1)) * (1 - x^(12*k-11))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 08 2015 *)

Formula

Expansion of 1 / f(-x, -x^11) in powers of x where f() is a Ramanujan theta function. - Michael Somos, Jan 10 2015
Partitions of n into parts of the form 12*k, 12*k+1, 12*k+11. - Michael Somos, Jan 10 2015
Euler transform of period 12 sequence [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, ...]. - Michael Somos, Jan 10 2015
G.f.: Product_{k>0} 1 / ((1 - x^(12*k)) * (1 - x^(12*k - 1)) * (1 - x^(12*k - 11))).
Convolution inverse of A247133.
a(n) ~ sqrt(2)*(1+sqrt(3)) * exp(Pi*sqrt(n/6)) / (8*n). - Vaclav Kotesovec, Nov 08 2015
a(n) = (1/n)*Sum_{k=1..n} A284372(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017
a(n) = a(n-1) + a(n-11) - a(n-14) - a(n-34) + + - - (with the convention a(n) = 0 for negative n), where 1, 11, 14, 34, ... is the sequence of generalized 14-gonal numbers A195818. - Peter Bala, Dec 10 2020

A214295 a(n) = 1 if n is a square, -1 if n is three times a square, 0 otherwise.

Original entry on oeis.org

1, 0, -1, 1, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0
Offset: 1

Views

Author

Michael Somos, Jul 10 2012

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
a(A092206(n)) = 0; a(A000290(n)) = 1; a(A033428(n)) = -1.

Examples

			G.f. = q - q^3 + q^4 + q^9 - q^12 + q^16 + q^25 - q^27 + q^36 - q^48 + q^49 + ...

Links

Crossrefs

Cf. A000700, A010052, A214293, A244612, A247133, A245485.

Programs

  • Haskell
    a214295 n = a010052 n - a010052 (3*n)  -- Reinhard Zumkeller, Jul 12 2012
  • Magma
    Basis( ModularForms( Gamma1(12), 1/2), 50) [2] ; /* Michael Somos, Jun 10 2014 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] - EllipticTheta[ 3, 0, q^3]) / 2, {q, 0, n}];
a[ n_] := Boole[ IntegerQ[ Sqrt[ n]]] - Boole[ IntegerQ[ Sqrt[ 3 n]]]; (* Michael Somos, Jun 10 2014 *)
Table[Which[IntegerQ[Sqrt[n]],1,IntegerQ[Sqrt[n/3]],-1,True,0],{n,120}] (* Harvey P. Dale, Apr 08 2013 *)
  • PARI
    {a(n) = issquare(n) - issquare(3*n)};
  • PARI
    {a(n) = if( n<1, 0, direuler( p=2, n, if( p==3, 1 - X, 1) / (1 - X^2 ))[n])};

Formula

Expansion of q * psi(q^3) * f(-q^2, -q^10) / f(-q^5, -q^7) in powers of q where psi(), f() are Ramanujan theta functions.
Multiplicative with a(3^e) = (-1)^e, a(p^e) = 1 if e even, 0 otherwise.
G.f.: (theta_3(q) - theta_3(q^3)) / 2 = Sum_{k>0} x^(k^2) - x^(3*k^2).
Dirichlet g.f.: zeta(2*s) * (1 - 3^(-s)). [corrected by Amiram Eldar, Oct 24 2023]
a(3*n) = - a(n). - Reinhard Zumkeller, Jul 12 2012
Expansion of (phi(q) - phi(q^3)) / 2 = q * chi(q) * f(-q, -q^11) in powers fof q where phi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Jan 10 2015
Euler transform of period 12 sequence [ 0, -1, 1, 0, 1, -1, 1, 0, 1, -1, 0, -1, ...]. - Michael Somos, Jan 10 2015
Convolution product of A000700 and A247133. - Michael Somos, Jan 10 2015
Sum_{k=1..n} a(k) ~ c*sqrt(n), where c = 1 - 1/sqrt(3) = 0.42264973... . - Amiram Eldar, Oct 24 2023

A133985 Expansion of f(-x, x^2) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, -1, 1, 0, 0, -1, 0, -1, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, 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, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Michael Somos, Oct 01 2007, Oct 04 2007

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
a(n) is nonzero if and only if n is a number of A001318.
The exponents in the q-series for this sequence are the squares of the numbers of A007310.
Number 14 of the 14 primitive eta-products which are holomorphic modular forms of weight 1/2 listed by D. Zagier on page 30 of "The 1-2-3 of Modular Forms". - Michael Somos, May 04 2016

Examples

			G.f. = 1 - x + x^2 - x^5 - x^7 + x^12 - x^15 + x^22 + x^26 - x^35 + x^40 + ...
G.f. = q - q^25 + q^49 - q^121 - q^169 + q^289 - q^361 + q^529 + q^625 + ...

Links

Crossrefs

Cf. A001318, A007310, A080995, A133988, A247133, A247223.

Programs

  • Mathematica
    a[ n_] := (-1)^n Boole[ IntegerQ[ Sqrt[24 n + 1]]]; (* Michael Somos, Jan 10 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^3]  QPochhammer[ x, -x], {x, 0, n}]; (* Michael Somos, Oct 30 2015 *)
  • PARI
    {a(n) = (-1)^n * issquare( 24*n + 1) };
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^5 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A))^2, n))};

Formula

Expansion of phi(x^3) / chi(x) in powers of x where phi(), chi() are Ramanujan theta functions.
Expansion of q^(-1/24) * eta(q) * eta(q^4) * eta(q^6)^5 / (eta(q^2) * eta(q^3) * eta(q^12))^2 in powers of q.
Euler transform of period 12 sequence [ -1, 1, 1, 0, -1, -2, -1, 0, 1, 1, -1, -1, ...].
a(n) = b(24*n + 1) where b() is multiplicative with b(p^(2*e)) = (-1)^e if p == 3, 5 (mod 8), b(p^(2*e)) = +1 if p == 1, 7 (mod 8) and b(p^(2*e-1)) = b(2^e) = b(3^e) = 0 if e>0.
G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = 4 (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A133988.
a(5*n + 3) = a(5*n + 4) = 0. a(25*n + 1) = -a(n). a(n) = (-1)^n * A080995(n).
G.f. Sum_{k>=0} a(k) * q^(24*k + 1) = Sum_{k in Z} (-1)^floor(k/2) * q^(6*k + 1)^2.
Expansion of f(-x^5, -x^7) - x * f(-x, -x^11) in powers of x. - Michael Somos, Jan 10 2015

A259660 Expansion of f(-x, -x^11) * psi(-x^3)^2 / psi(-x) in powers of x where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 0, 0, -1, 1, 1, 0, 0, 1, 0, 0, -1, 1, 0, 0, 1, 1, -1, 0, -1, 2, 1, 0, 0, 0, 1, 0, -1, 1, 0, 0, -1, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, -1, 1, 0, 0, 0, 1, -1, 0, -1, 0, 1, 0, 2, 1, 0, 0, -2, 2, 0, 0, 0, 1, 1, 0, -1, 0, 1, 0, 0, 2, -1, 0, 0, 1, 0, 0, 0, 1, -1
Offset: 0

Views

Author

Michael Somos, Jul 02 2015

Keywords

Comments

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

Examples

			G.f. = 1 - x^3 + x^4 + x^5 + x^8 - x^11 + x^12 + x^15 + x^16 - x^17 - x^19 + ...
G.f. = q^5 - q^14 + q^17 + q^20 + q^29 - q^38 + q^41 + q^50 + q^53 - q^56 + ...

Links

Crossrefs

Cf. A121444, A247133, A259529.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{ 0, 2, 0, 0, 1, -1, -1, 0, -1, -1, 1, 0}[[Mod[ k, 12, 1]]], {k, n}], {x, 0, n}];
QP:= QPochhammer; a[n_]:= SeriesCoefficient[(QP[x, x^12]*QP[x^11,x^12]* QP[x^12]*QP[x^3, -x^3]^2*QP[x^6]^2)/(QP[x, -x]*QP[x^2]), {x, 0, n}]; Table[a[n], {n, 0, 100}] (* G. C. Greubel, Mar 17 2018 *)
  • PARI
    {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k)^([ 2, 0, 0, 1, -1, -1, 0, -1, -1, 1, 0, 0][k%12 + 1]), 1 + x * O(x^n)), n))};

Formula

Expansion of f(-x, -x^11) * f(x, x^5)^2 / f(x) in powers of x where f(,) is the Ramanujan general theta function.
Euler transform of period 12 sequence [ 0, 0, -1, 1, 1, 0, 1, 1, -1, 0, 0, -2, ...].
a(4*n) = A121444(n). a(4*n + 1) = a(n - 1). a(4*n + 2) = 0.
Convolution of A247133 and A259529.

A287325 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Sum_{j=-inf..inf} (-1)^j*x^(k*j*(j-1)/2 + j^2).

Original entry on oeis.org

1, 1, -2, 1, -1, 0, 1, -1, -1, 0, 1, -1, 0, 0, 2, 1, -1, 0, -1, 0, 0, 1, -1, 0, 0, 0, 1, 0, 1, -1, 0, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 1, 1, 0, 1, -1, 0, 0, 0, -1, 0, 0, 0, -2, 1, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 13 2017

Keywords

Examples

			Square array begins:
   1,   1,   1,   1,   1,   1, ...
  -2,  -1,  -1,  -1,  -1,  -1, ...
   0,  -1,   0,   0,   0,   0, ...
   0,   0,  -1,   0,   0,   0, ...
   2,   0,   0,  -1,   0,   0, ...
   0,   1,   0,   0,  -1,   0, ...

Links

Crossrefs

Columns k=0..10 give A002448, A010815, A106459, A113429, A089802, A232714, A244525, A281814 (absolute values), A205988 (absolute values), A281815 (absolute values), A247133.
Cf. A000041, A000290, A000326, A000384, A000566, A000567, A001106, A001107, A006950, A015128, A036820, A051682, A051624, A051865, A195825, A195848, A195849, A195850, A195851, A195852, A196933, A211970.

Programs

  • Mathematica
    Table[Function[k, SeriesCoefficient[Sum[(-1)^i x^(k i (i - 1)/2 + i^2), {i, -n, n}], {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[Product[(1 - x^((k + 2) i)) (1 - x^((k + 2) i - 1)) (1 - x^((k + 2) i - k - 1)), {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[(x^(2 + k) QPochhammer[1/x, x^(2 + k)] QPochhammer[x^(-1 - k), x^(2 + k)] QPochhammer[x^(2 + k), x^(2 + k)])/((-1 + x) (-1 + x^(1 + k))), {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten

Formula

G.f. of column 0: Sum_{j=-inf..inf} (-1)^j*x^A000290(j) = Product_{i>=1} (1 + x^i)/(1 - x^i) (convolution inverse of A015128).
G.f. of column 1: Sum_{j=-inf..inf} (-1)^j*x^A000326(j) = Product_{i>=1} (1 - x^i) (convolution inverse of A000041).
G.f. of column 2: Sum_{j=-inf..inf} (-1)^j*x^A000384(j) = Product_{i>=1} (1 - x^(2*i))/(1 + x^(2*i-1)) (convolution inverse of A006950).
G.f. of column 3: Sum_{j=-inf..inf} (-1)^j*x^A000566(j) = Product_{i>=1} (1 - x^(5*i))*(1 - x^(5*i-1))*(1 - x^(5*i-4)) (convolution inverse of A036820).
G.f. of column 4: Sum_{j=-inf..inf} (-1)^j*x^A000567(j) = Product_{i>=1} (1 - x^(6*i))*(1 - x^(6*i-1))*(1 - x^(6*i-5)) (convolution inverse of A195848).
G.f. of column 5: Sum_{j=-inf..inf} (-1)^j*x^A001106(j) = Product_{i>=1} (1 - x^(7*i))*(1 - x^(7*i-1))*(1 - x^(7*i-6)) (convolution inverse of A195849).
G.f. of column 6: Sum_{j=-inf..inf} (-1)^j*x^A001107(j) = Product_{i>=1} (1 - x^(8*i))*(1 - x^(8*i-1))*(1 - x^(8*i-7)) (convolution inverse of A195850).
G.f. of column 7: Sum_{j=-inf..inf} (-1)^j*x^A051682(j) = Product_{i>=1} (1 - x^(9*i))*(1 - x^(9*i-1))*(1 - x^(9*i-8)) (convolution inverse of A195851).
G.f. of column 8: Sum_{j=-inf..inf} (-1)^j*x^A051624(j) = Product_{i>=1} (1 - x^(10*i))*(1 - x^(10*i-1))*(1 - x^(10*i-9)) (convolution inverse of A195852).
G.f. of column 9: Sum_{j=-inf..inf} (-1)^j*x^A051865(j) = Product_{i>=1} (1 - x^(11*i))*(1 - x^(11*i-1))*(1 - x^(11*i-10)) (convolution inverse of A196933).
G.f. of column k: Sum_{j=-inf..inf} (-1)^j*x^(k*j*(j-1)/2+j^2) = Product_{i>=1} (1 - x^((k+2)*i))*(1 - x^((k+2)*i-1))*(1 - x^((k+2)*i-k-1)).
Showing 1-6 of 6 results.

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