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.

Previous Showing 11-18 of 18 results.

A294408 Expansion of 1/(1 + Sum_{i>=1} q^(i^2)/Product_{j=1..i} (1 + q^(2*j))).

Original entry on oeis.org

1, -1, 1, 0, -2, 3, -2, -1, 6, -10, 8, 2, -19, 34, -30, -3, 60, -112, 106, -2, -188, 370, -373, 48, 586, -1226, 1307, -296, -1808, 4046, -4546, 1430, 5516, -13300, 15724, -6217, -16626, 43566, -54132, 25464, 49373, -142146, 185496, -100306, -143896, 461874, -632864, 384348, 409270, -1494356, 2150240
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 30 2017

Keywords

Comments

Convolution inverse of the 3rd order mock theta function phi(q) (A053250).

Links

Crossrefs

Cf. A053250, A081360, A294407.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[1/(1 + Sum[q^(i^2)/Product[1 + q^(2 j), {j, 1, i}], {i, 1, nmax}]), {q, 0, nmax}], q]

Formula

G.f.: 1/(1 + Sum_{i>=1} q^(i^2)/Product_{j=1..i} (1 + q^(2*j))).

A027357 Number of partitions of n into distinct odd parts, the greatest being congruent to 1 mod 4.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 2, 1, 2, 4, 3, 2, 3, 5, 5, 3, 5, 8, 6, 5, 7, 10, 10, 8, 11, 15, 14, 12, 15, 20, 20, 18, 22, 28, 27, 25, 30, 37, 38, 35, 42, 51, 50, 49, 56, 67, 69, 67, 77, 90, 91, 90, 102, 117, 121, 121, 135, 155, 159, 160, 177
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

Also number of self-conjugate partitions of n into an odd number of parts. - Vladeta Jovovic, Feb 18 2004

Crossrefs

Cf. A027356, A027358.
Cf. A000700, A053250.

Formula

a(n) = (A000700(n)-(-1)^n*A053250(n))/2. - Vladeta Jovovic, Mar 12 2006
a(n) = Sum_{k=0..floor(n/4)-1} A027356(n, 4*k+1). - Sean A. Irvine, Oct 28 2019

A027358 Number of partitions of n into distinct odd parts, the greatest being congruent to 3 mod 4.

Original entry on oeis.org

0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 2, 2, 1, 1, 3, 3, 1, 2, 4, 4, 3, 3, 6, 6, 4, 6, 9, 9, 7, 8, 12, 12, 10, 12, 17, 18, 15, 17, 23, 24, 21, 25, 32, 33, 31, 34, 43, 45, 42, 48, 58, 61, 58, 64, 77, 80, 78, 87, 102, 107, 106, 115, 134, 141, 139, 153, 175
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

Also number of self-conjugate partitions of n into an even number of parts. - Vladeta Jovovic, Feb 18 2004

Crossrefs

Cf. A027356, A027357.
Cf. A000700, A053250.

Formula

a(n) = (A000700(n)+(-1)^n*A053250(n))/2. - Vladeta Jovovic, Mar 12 2006
a(n) + A027357(n) = A000700(n). - R. J. Mathar, Oct 03 2016
a(n) = Sum_{k=0..floor(n/4)-1} A027356(n, 4*k+3). - Sean A. Irvine, Oct 28 2019

A257657 Expansion of f(-x, -x) * f(-x^6, -x^6) / f(x, x^2) in powers of x where f(,) is Ramanujan's general theta function.

Original entry on oeis.org

1, -3, 2, 1, -1, -1, 3, -1, 0, -2, -2, 2, 1, -3, 3, 4, -1, -3, 1, 0, -1, -2, 0, 3, 1, -6, 2, 4, -4, -1, 4, 2, -1, -3, 0, 5, -1, -9, 5, 7, -4, -7, 4, 5, -3, -4, 0, 8, -1, -13, 4, 11, -7, -7, 7, 6, -1, -10, 0, 14, -1, -15, 8, 15, -10, -14, 8, 11, -7, -13, 2, 17
Offset: 0

Views

Author

Michael Somos, Jul 26 2015

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

Links

Crossrefs

Cf. A053250, A260413.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2]^2 QPochhammer[ x] QPochhammer[ -x^3] / QPochhammer[ x^3], {x, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^6 + A)^3 / (eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^12 + A)), n))};

Formula

Expansion of q^(1/24) * eta(q)^3 * eta(q^6)^3 / (eta(q^2)^2 * eta(q^3)^2 * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ -3, -1, -1, -1, -3, -2, -3, -1, -1, -1, -3, -1, ...].
a(n) = 2 * A260413(n) - A053250(n).

A261401 Expansion of 1 + Sum_{n>=1} ( q^(n^2)/Product_{i=1..n} ((1+q^(2*i))^2) ).

Original entry on oeis.org

1, 1, 0, -2, 1, 3, -2, -4, 1, 6, 0, -8, 2, 8, -4, -10, 3, 15, -2, -16, 4, 16, -8, -22, 7, 28, -2, -30, 7, 33, -14, -40, 9, 48, -8, -54, 16, 60, -20, -68, 19, 82, -16, -94, 21, 98, -34, -116, 32, 138, -24, -146, 40, 165, -54, -190, 45, 212, -46, -240
Offset: 0

Views

Author

N. J. A. Sloane, Aug 20 2015

Keywords

Crossrefs

Suggested by the g.f.'s for A000025 and A053250.

Programs

  • Maple
    phi:=1+add( q^(n^2)/mul((1+q^(2*i))^2,i=1..n), n=1..61):
series(phi,q,60);
seriestolist(%);

A109471 Cumulative sum of absolute values of coefficients of q^(2n) in the series expansion of Ramanujan's mock theta function f(q).

Original entry on oeis.org

1, 3, 6, 11, 17, 27, 38, 55, 76, 103, 136, 182, 235, 303, 385, 489, 612, 766, 945, 1166, 1428, 1742, 2111, 2557, 3072, 3686, 4401, 5246, 6223, 7371, 8692, 10236, 12014, 14074, 16435, 19171, 22292, 25884, 29981, 34677, 40017, 46122, 53038, 60920
Offset: 0

Views

Author

Jonathan Vos Post, Aug 28 2005

Keywords

Links

Crossrefs

Cf. A000039, A000019, A053250, A053251, A053252, A053253, A053254, A053255, A064053.

Programs

  • Mathematica
    nmax = 200; f[q_, s_] := Sum[q^(n^2)/Product[1 + q^k, {k, n}]^2, {n, 0, s}]; A000039:= CoefficientList[Series[f[q, nmax], {q, 0, nmax}], q][[1 ;; -1 ;; 2]]; Table[Sum[Abs[A000039[[k]]], {k,1,n}], {n,1,51}] (* G. C. Greubel, Feb 18 2018 *)

Formula

a(n) = Sum_{k=0..n} abs(A000039(k)). [corrected by Joerg Arndt, Feb 25 2018]
a(n) ~ sqrt(3/2) * exp(sqrt(n/3)*Pi) / Pi. - Vaclav Kotesovec, Jun 12 2019

A133738 Expansion of product of 3rd order mock theta function phi(q) and Ramanujan theta function f(-q) in powers of q.

Original entry on oeis.org

1, 0, -2, -2, 2, 2, -2, 0, 2, 4, -2, -4, 2, 0, -2, -2, 2, 4, -4, -4, 2, 2, -2, 0, 4, 4, 0, -6, 2, 0, -2, 0, 2, 6, -4, -4, 4, 0, -4, -2, 0, 4, -2, -4, 2, 0, 0, 0, 4, 4, -2, -6, 2, 0, -6, 2, 2, 8, 0, -4, 2, 0, 0, 0, 2, 2, -6, -4, 2, 0, -2, 0, 4, 4, 0, -6, 2, -2
Offset: 0

Views

Author

Michael Somos, Sep 22 2007

Keywords

Comments

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

Examples

			G.f. = 1 - 2*x^2 - 2*x^3 + 2*x^4 + 2*x^5 - 2*x^6 + 2*x^8 + 4*x^9 - 2*x^10 + ...

Links

Crossrefs

Cf. A010815, A053250.

Programs

  • Mathematica
    a[ n_] := If[ n < 0, 0, With[ {m = Quotient[ Sqrt[24 n + 1] - 1, 6]}, SeriesCoefficient[ Sum[ (-1)^k x^(k (3 k + 1)/2) (1 + x^k) / (1 + x^(2 k)), {k, -m, m}], {x, 0, n}]]]; (* Michael Somos, Jul 26 2015 *)
  • PARI
    {a(n) = if( n<0, 0, polcoeff( 1 + 2 * sum(k=1, (sqrtint(24*n + 1) - 1) \ 6, (-1)^k * x^(k*(3*k + 1)/2) * (1 + x^k) / (1 + x^(2*k)), x * O(x^n)), n))};
  • PARI
    {a(n) = my(t); if( n<0, 0, t = 1 + O(x^n); polcoeff( sum(k=1, sqrtint(n), t *= x^(2*k - 1) / (1 + x^(2*k)) + x * O(x^(n - (k-1)^2)), 1) * eta(x + x * O(x^n)), n))};

Formula

G.f.: Sum_{k in Z} (-1)^k * x^(k*(3*k + 1)/2) * (1 + x^k) / (1 + x^(2*k)).
G.f.: ( Product_{k>0} 1 - x^k ) * ( 1 + Sum_{k>0} x^k^2 / ((1 + x^2) * (1 + x^4) * ... * (1 + x^(2*k))) ).
Convolution of A053250 and A010815.

A260460 Expansion of f(-q) in powers of q where f() is a 3rd order mock theta function.

Original entry on oeis.org

1, -1, -2, -3, -3, -3, -5, -7, -6, -6, -10, -12, -11, -13, -17, -20, -21, -21, -27, -34, -33, -36, -46, -51, -53, -58, -68, -78, -82, -89, -104, -118, -123, -131, -154, -171, -179, -197, -221, -245, -262, -279, -314, -349, -369, -398, -446, -486, -515, -557
Offset: 0

Views

Author

Michael Somos, Jul 26 2015

Keywords

Examples

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

Crossrefs

Cf. A000025, A053250, A053251, A132969.

Programs

  • Mathematica
    a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ (-x)^k^2 / Product[ 1 + (-x)^j, {j, k}]^2, {k, 0, Sqrt@n}], {x, 0, n}]];
  • PARI
    {a(n) = my(t); if( n<0, 0, t = 1 + O(x^n); polcoeff( sum(k=1, sqrtint(n), t *= (-x)^(2*k - 1) / (1 + (-x)^k)^2 + x * O(x^(n - (k-1)^2)), 1), n))};

Formula

G.f.: Sum_{k>=0} (-x)^(k^2) / Product_{i=1..k} (1 + (-x)^i)^2.
G.f.: 2 * (Sum_{k in Z} (-1)^k * x^(k*(3*k + 1)/2) / (1 + x^k)) / (Sum_{k in Z} (-1)^k * x^(k*(3*k + 1)/2))
a(n) = (-1)^n * A000025(n). a(n) < 0 if n>0.
a(n) = A053250(n) - 2 * A053251(n) = 2 * A053250(n) - A132969(n) = A132969(n) - 4 * A053251(n).
Previous Showing 11-18 of 18 results.

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