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 1-4 of 4 results.

A132970 Expansion of phi(-x) * chi(-x) in powers of x where phi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

1, -3, 2, -1, 5, -5, 3, -5, 6, -10, 10, -8, 13, -15, 15, -16, 23, -27, 25, -30, 35, -40, 42, -45, 55, -66, 68, -70, 86, -95, 100, -110, 125, -141, 150, -161, 185, -207, 215, -235, 266, -293, 310, -335, 375, -410, 438, -470, 521, -575, 610, -653, 725, -785, 835, -900, 983, -1070, 1140, -1220, 1331
Offset: 0

Views

Author

Michael Somos, Sep 04 2007

Keywords

Comments

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

Examples

			G.f. = 1 - 3*x + 2*x^2 - x^3 + 5*x^4 - 5*x^5 + 3*x^6 - 5*x^7 + 6*x^8 + ...
G.f. = 1/q - 3*q^23 + 2*q^47 - q^71 + 5*q^95 - 5*q^119 + 3*q^143 - 5*q^167 + ...

References

  • N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 60, Eqs. (26.64),(26.65),(26.66)

Links

Crossrefs

CF. A124226, A132969.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] QPochhammer[ x, x^2], {x, 0, n}]; (* Michael Somos, Jul 20 2015 *)
  • PARI
    {a(n) = if( n<0, 0, polcoeff( prod(k=1, (n+1)\2, 1 - x^(2*k-1), 1 + x * O(x^n)) * sum(k=1, sqrtint(n), 2 * (-1)^k * x^k^2, 1), n))};
  • PARI
    {a(n) = my(A) ; if( n<0, 0, A = x * O(x^n) ; polcoeff( eta(x + A)^3 / eta(x^2 + A)^2, n))};

Formula

Expansion of phi(-q) + 2 * psi(-q) in powers of q where phi(), psi() are Ramanujan 3rd order mock theta functions.
Expansion of q^(1/24) * eta(q)^3 / eta(q^2)^2 in powers of q.
Euler transform of period 2 sequence [ -3, -1, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = 48^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A085140.
G.f.: ( Sum_{k in Z} (-1)^k * x^k^2 ) / ( Product_{k>0} (1 + x^k) ).
G.f.: Product_{k>0} (1 - x^k) / (1 + x^k)^2.
a(n) = (-1)^n * A132969(n). a(n) = A124226(n) unless n=1.
a(n) ~ (-1)^n * exp(Pi*sqrt(n/6)) / (2*sqrt(n)). - Vaclav Kotesovec, Oct 14 2017

A132969 Expansion of phi(q) * chi(q) in powers of q where phi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

1, 3, 2, 1, 5, 5, 3, 5, 6, 10, 10, 8, 13, 15, 15, 16, 23, 27, 25, 30, 35, 40, 42, 45, 55, 66, 68, 70, 86, 95, 100, 110, 125, 141, 150, 161, 185, 207, 215, 235, 266, 293, 310, 335, 375, 410, 438, 470, 521, 575, 610, 653, 725, 785, 835, 900, 983, 1070, 1140
Offset: 0

Views

Author

Michael Somos, Sep 04 2007

Keywords

Comments

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

Examples

			G.f. = 1 + 3*x + 2*x^2 + x^3 + 5*x^4 + 5*x^5 + 3*x^6 + 5*x^7 + 6*x^8 + 10*x^9 + ...
G.f. = 1/q + 3*q^23 + 2*q^47 + q^71 + 5*q^95 + 5*q^119 + 3*q^143 + 5*q^167 +...

References

  • N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; top of p. 60.

Links

Crossrefs

Cf. A053250, A053251, A124226, A132970.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[Product[(1 - x^(2*k)) * ( (1 + x^k) / (1 + x^(2*k)) )^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x, x^2], {x, 0, n}]; (* Michael Somos, Oct 31 2015 *)
  • PARI
    {a(n) = if( n<0, 0, polcoeff( prod(k=1, (n+1)\2, 1 + x^(2*k-1), 1 + x*O(x^n)) * sum(k=1, sqrtint(n), 2 * x^k^2, 1), n))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 / (eta(x + A) * eta(x^4 + A))^3, n))};

Formula

Expansion of phi(q) + 2 * psi(q) in powers of q where phi(), psi() are Ramanujan 3rd order mock theta functions.
Expansion of q^(1/24) * eta(q^2)^7 / (eta(q) * eta(q^4))^3 in powers of q.
Euler transform of period 4 sequence [ 3, -4, 3, -1, ...].
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).
G.f.: ( Sum_{k in Z} x^k^2 ) * ( Product_{k>0} (1 + x^(2*k-1)) ).
G.f.: Product_{k>0} (1 - x^(2*k)) * ((1 + x^k) / (1 + x^(2*k)))^3.
a(n) = (-1)^n * A132970(n). a(n) = (-1)^n * A124226(n) unless n=1.
a(n) ~ exp(Pi*sqrt(n/6)) / (2*sqrt(n)). - Vaclav Kotesovec, Sep 08 2015

A124227 Number of partitions of n with even crank.

Original entry on oeis.org

1, 0, 2, 1, 5, 1, 7, 5, 14, 10, 26, 24, 45, 43, 75, 80, 127, 135, 205, 230, 331, 376, 522, 605, 815, 946, 1252, 1470, 1902, 2235, 2852, 3366, 4237, 5001, 6230, 7361, 9081, 10715, 13115, 15475, 18802, 22145, 26742, 31463, 37775, 44362, 52998, 62142
Offset: 0

Views

Author

Vladeta Jovovic, Oct 20 2006

Keywords

Comments

For a partition p, let l(p) = largest part of p, w(p) = number of 1's in p, m(p) = number of parts of p larger than w(p). The crank of p is given by l(p) if w(p) = 0, otherwise m(p)-w(p).

Crossrefs

Cf. A000041, A124226, A124228.

Programs

  • Maple
    A000041 := proc(n) combinat[numbpart](n) ; end: A124226 := proc(n) local x,gf,i ; gf := 1; for i from 1 to n+1 do gf := taylor(gf*(1-x^i)/(1+x^i)^2,x=0,n+1) ; od ; coeftayl(2*x+gf,x=0,n) ; end: A124227 := proc(n) (A000041(n)+A124226(n))/2 ; end: for n from 0 to 60 do printf("%a, ",A124227(n)) ; od ; # R. J. Mathar, May 18 2007
  • Mathematica
    A132970[n_] := SeriesCoefficient[EllipticTheta[4, 0, x] QPochhammer[x, x^2], {x, 0, n}];
a[n_] := If[n == 1, 0, (PartitionsP[n] + A132970[n])/2];
Table[a[n], {n, 0, 47}] (* Jean-François Alcover, Oct 26 2023, after Michael Somos in A124226 *)

Formula

a(n) = (A000041(n) + A124226(n))/2.

Extensions

More terms from R. J. Mathar, May 18 2007

A124228 Number of partitions of n with odd crank.

Original entry on oeis.org

0, 1, 0, 2, 0, 6, 4, 10, 8, 20, 16, 32, 32, 58, 60, 96, 104, 162, 180, 260, 296, 416, 480, 650, 760, 1012, 1184, 1540, 1816, 2330, 2752, 3476, 4112, 5142, 6080, 7522, 8896, 10922, 12900, 15710, 18536, 22438, 26432, 31798, 37400, 44772, 52560, 62612
Offset: 0

Views

Author

Vladeta Jovovic, Oct 20 2006

Keywords

Comments

For a partition p, let l(p) = largest part of p, w(p) = number of 1's in p, m(p) = number of parts of p larger than w(p). The crank of p is given by l(p) if w(p) = 0, otherwise m(p)-w(p).

Crossrefs

Cf. A000041, A124226, A124227.

Programs

  • Maple
    A000041 := proc(n) combinat[numbpart](n) ; end: A124226 := proc(n) local x,gf,i ; gf := 1; for i from 1 to n+1 do gf := taylor(gf*(1-x^i)/(1+x^i)^2,x=0,n+1) ; od ; coeftayl(2*x+gf,x=0,n) ; end: A124228 := proc(n) (A000041(n)-A124226(n))/2 ; end: for n from 0 to 60 do printf("%a, ",A124228(n)) ; od ; # R. J. Mathar, May 18 2007
  • Mathematica
    A132970[n_] := SeriesCoefficient[EllipticTheta[4, 0, x] QPochhammer[x, x^2], {x, 0, n}];
a[n_] := If[n < 2, n, (PartitionsP[n] - A132970[n])/2];
Table[a[n], {n, 0, 47}] (* Jean-François Alcover, Oct 26 2023, after Michael Somos in A124226 *)

Formula

a(n) = (A000041(n)-A124226(n))/2.

Extensions

More terms from R. J. Mathar, May 18 2007
Showing 1-4 of 4 results.

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