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 1-8 of 8 results.

A278399 G.f.: Re((i; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

Original entry on oeis.org

1, -1, -1, -2, -2, -3, -2, -3, -2, -2, 0, 0, 3, 4, 8, 9, 14, 16, 22, 24, 30, 32, 39, 40, 46, 46, 51, 48, 52, 46, 46, 36, 32, 16, 8, -15, -30, -60, -82, -122, -151, -200, -238, -296, -342, -412, -464, -542, -602, -686, -750, -841, -904, -996, -1058, -1146, -1198
Offset: 0

Views

Author

Vladimir Reshetnikov, Nov 20 2016

Keywords

Comments

The q-Pochhammer symbol (a; q)inf = Product{k>=0} (1 - a*q^k).

Links

Crossrefs

Cf. A278400, A278401, A278402, A278420, A292042, A292043.

Programs

  • Maple
    G := add((-1)^n*x^(n*(2*n-1))/mul(1 - x^k, k = 1..2*n), n = 0..7):
S := series(G, x, 101):
seq(coeff(S, x, j), j = 0..100); # Peter Bala, Feb 04 2021
  • Mathematica
    Re[(QPochhammer[I, x] + O[x]^60)[[3]]]

Formula

(i; x)_inf is the g.f. for a(n) + i*A278400(n).
G.f.: Sum_{n >= 0} (-1)^n^x^(n*(2*n-1))/Product_{k = 1..2*n} 1 - x^k. - Peter Bala, Feb 04 2021

A278400 G.f.: Im((i; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

Original entry on oeis.org

-1, -1, -1, 0, 0, 1, 2, 3, 4, 6, 6, 8, 9, 10, 10, 11, 10, 10, 8, 6, 2, 0, -7, -12, -20, -28, -39, -48, -62, -74, -90, -104, -122, -136, -156, -171, -190, -204, -222, -232, -247, -252, -260, -258, -258, -244, -232, -204, -176, -130, -84, -15, 54, 148, 244, 368
Offset: 0

Views

Author

Vladimir Reshetnikov, Nov 20 2016

Keywords

Comments

The q-Pochhammer symbol (a; q)inf = Product{k>=0} (1 - a*q^k).

Links

Crossrefs

Cf. A278399, A278401, A278402, A278420, A292042, A292043.

Programs

  • Maple
    with(gfun): series(add((-1)^(n+1)*x^(n*(2*n+1))/mul(1 - x^k, k = 1..2*n+1), n = 0..6), x, 100): seriestolist(%); # Peter Bala, Feb 06 2021
  • Mathematica
    Im[(QPochhammer[I, x] + O[x]^60)[[3]]]

Formula

(i; x)_inf is the g.f. for A278399(n) + i*a(n).
G.f.: Sum_{n >= 0} (-1)^(n+1)*x^(n*(2*n+1))/Product_{k = 1..2*n+1} 1 - x^k. - Peter Bala, Feb 06 2021

A292043 G.f.: Im((i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

Original entry on oeis.org

0, -1, -1, -1, -1, -1, 0, 0, 1, 2, 3, 4, 6, 7, 9, 10, 12, 13, 15, 15, 16, 16, 16, 14, 13, 9, 6, 0, -5, -14, -22, -34, -45, -60, -74, -93, -110, -132, -152, -177, -199, -226, -249, -277, -300, -328, -348, -373, -389, -408, -417, -428, -425, -424, -407, -389, -352
Offset: 0

Views

Author

Seiichi Manyama, Sep 08 2017

Keywords

Examples

			Product_{k>=1} (1 - i*x^k) = 1 + (0-1i)*x + (0-1i)*x^2 + (-1-1i)*x^3 + (-1-1i)*x^4 + (-2-1i)*x^5 + (-2+0i)*x^6 + (-3+0i)*x^7 + ...

Links

Crossrefs

Cf. A278399, A278400, A278420, A292042, A292052.

Programs

  • Maple
    N:= 100:
S := convert(series( add( (-1)^(n+1)*x^((n+1)*(2*n+1))/(mul(1 - x^k,k = 1..2*n+1)), n = 0..floor(sqrt(N/2)) ), x, N+1 ), polynom):
seq(coeff(S, x, n), n = 0..N); # Peter Bala, Feb 05 2021
  • Mathematica
    Im[CoefficientList[Series[QPochhammer[I*x, x], {x, 0, 100}], x]] (* Vaclav Kotesovec, Sep 08 2017 *)

Formula

(i*x; x)_inf is the g.f. for A292042(n) + i*a(n).
A292042(n)^2 + a(n)^2 = A278420(n). - Vaclav Kotesovec, Sep 08 2017
From Peter Bala, Feb 05 2021: (Start)
G.f.: Sum_{n >= 0} (-1)^(n+1)*x^((n+1)*(2*n+1))/Product_{k = 1..2*n+1} (1 - x^k).
The 2 X 2 matrix Product_{k >= 1} [1, -x^k; x^k, 1] = [A(x), B(x); -B(x), A(x)], where A(x) is the g.f. of A292042 and B(x) is the g.f. for this sequence.
A(x)^2 + B(x)^2 = Product_{k >= 1} 1 + x^(2*k) = A000009(x^2).
A(x) + B(x) is the g.f. of A278399; B(x) - A(x) is the g.f. of A278400. (End)

A278420 a(n) = (A278399(n)^2 + A278400(n)^2)/2.

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 4, 9, 10, 20, 18, 32, 45, 58, 82, 101, 148, 178, 274, 306, 452, 512, 785, 872, 1258, 1450, 2061, 2304, 3274, 3796, 5108, 6056, 7954, 9376, 12200, 14733, 18500, 22608, 28004, 34354, 41905, 51752, 62122, 77090, 91764, 114640, 134560, 167690
Offset: 0

Views

Author

Vladimir Reshetnikov, Nov 21 2016

Keywords

Links

Crossrefs

Cf. A278399, A278400, A292042, A292043.

Programs

  • Mathematica
    Abs[(QPochhammer[I, x] + O[x]^60)[[3]]]^2 / 2

Formula

a(n) = A292042(n)^2 + A292043(n)^2. - Vaclav Kotesovec, Sep 08 2017

A292137 G.f.: Im(1/(i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

Original entry on oeis.org

0, 1, 1, 0, 0, 0, -1, -2, -2, -2, -2, -3, -3, -2, -2, -2, -1, 1, 2, 2, 4, 6, 7, 8, 10, 13, 14, 14, 15, 17, 17, 15, 15, 16, 14, 10, 8, 6, 1, -5, -10, -14, -21, -31, -38, -43, -53, -64, -71, -77, -86, -97, -104, -108, -115, -124, -127, -125, -127, -130, -125, -116
Offset: 0

Views

Author

Seiichi Manyama, Sep 09 2017

Keywords

Examples

			Product_{k>=1} 1/(1 - i*x^k) = 1 + (0+1i)*x + (-1+1i)*x^2 + (-1+0i)*x^3 + (-1+0i)*x^4 + (-1+0i)*x^5 + (-2-1i)*x^6 + (-1-2i)*x^7 + ...

Links

Crossrefs

Cf. A000108, A010815, A027193, A063834, A067659, A081362, A099323, A196545, A220418, A290261, A292042, A292043, A292136, A292138, A298118, A300355.

Programs

  • Maple
    N:= 100:
S := convert(series( add( (-1)^n*x^(2*n+1)/(mul(1 - x^k,k = 1..2*n+1)), n = 0..N ), x, N+1 ), polynom):
seq(coeff(S, x, n), n = 0..N); # Peter Bala, Jan 15 2021
  • Mathematica
    Im[CoefficientList[Series[1/QPochhammer[I*x, x], {x, 0, 100}], x]] (* Vaclav Kotesovec, Sep 17 2017 *)

Formula

1/(i*x; x)_inf is the g.f. for A292136(n) + i*a(n).
a(n) = Sum (-1)^((k - 1)/2) where the sum is over all integer partitions of n into an odd number of parts and k is the number of parts. - Gus Wiseman, Mar 08 2018
G.f.: Sum_{n >= 0} (-1)^n * x^(2*n+1)/Product_{k = 1..2*n+1} (1 - x^k). - Peter Bala, Jan 15 2021

A292136 G.f.: Re(1/(i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

Original entry on oeis.org

1, 0, -1, -1, -1, -1, -2, -1, 0, 0, 0, 1, 2, 3, 3, 4, 6, 6, 5, 6, 7, 6, 5, 5, 5, 3, 0, -2, -3, -6, -11, -13, -14, -19, -24, -27, -29, -33, -38, -40, -40, -43, -47, -46, -43, -43, -43, -38, -30, -26, -22, -12, 1, 11, 20, 36, 56, 71, 85, 106, 130, 149, 166, 190, 217
Offset: 0

Views

Author

Seiichi Manyama, Sep 09 2017

Keywords

Examples

			Product_{k>=1} 1/(1 - i*x^k) = 1 + (0+1i)*x + (-1+1i)*x^2 + (-1+0i)*x^3 + (-1+0i)*x^4 + (-1+0i)*x^5 + (-2-1i)*x^6 + (-1-2i)*x^7 + ...
Product_{k>=1} 1/(1 + i*x^k) = 1 + (0-1i)*x + (-1-1i)*x^2 + (-1+0i)*x^3 + (-1+0i)*x^4 + (-1+0i)*x^5 + (-2+1i)*x^6 + (-1+2i)*x^7 + ...
From _Peter Bala_, Jan 19 2021: (Start)
The number of partitions of n = 13 into an even number of parts is:
# parts (2*k)   2    4    6   8   10   12
# partitions    6   18   14   7    3    1
Hence a(13) = Sum (-1)^k = -6 + 18 - 14 + 7 - 3 + 1 = 3. (End)

Links

Crossrefs

Cf. A292042, A292043, A292137, A292138.

Programs

  • Maple
    N:= 100:
S := convert(series( add( (-1)^n*x^(2*n)/(mul(1 - x^k,k = 1..2*n)), n = 0..N ), x, N+1 ), polynom):
seq(coeff(S, x, n), n = 0..N); # Peter Bala, Jan 15 2021
  • Mathematica
    Re[CoefficientList[Series[1/QPochhammer[I*x, x], {x, 0, 100}], x]] (* Vaclav Kotesovec, Sep 17 2017 *)

Formula

1/( i*x; x)_inf is the g.f. for a(n) + i*A292137(n).
1/(-i*x; x)_inf is the g.f. for a(n) + i*A292138(n).
From Peter Bala, Jan 19 2021: (Start)
a(n) = Sum (-1)^k, where the sum is over all integer partitions of n into an even number of parts and 2*k is the number of parts in a partition. An example is given below.
G.f.: Sum_{n >= 0} (-1)^n * x^(2*n)/Product_{k = 1..2*n} (1 - x^k). (End)

A292052 G.f.: Im((-i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 0, 0, -1, -2, -3, -4, -6, -7, -9, -10, -12, -13, -15, -15, -16, -16, -16, -14, -13, -9, -6, 0, 5, 14, 22, 34, 45, 60, 74, 93, 110, 132, 152, 177, 199, 226, 249, 277, 300, 328, 348, 373, 389, 408, 417, 428, 425, 424, 407, 389, 352, 314, 252
Offset: 0

Views

Author

Seiichi Manyama, Sep 08 2017

Keywords

Examples

			Product_{k>=1} (1 + i*x^k) = 1 + (0+1i)*x + (0+1i)*x^2 + (-1+1i)*x^3 + (-1+1i)*x^4 + (-2+1i)*x^5 + (-2+0i)*x^6 + (-3+0i)*x^7 + ...

Links

Crossrefs

Cf. A278399, A278400, A278420, A292042, A292043.

Programs

  • Maple
    N:= 100: # to get a(0)..a(N)
P:= mul(1+I*x^k, k=1..N):
S:= series(P, x, N+1):
seq(evalc(Im(coeff(S,x,j))),j=0..N); # Robert Israel, Sep 08 2017
  • Mathematica
    Im[CoefficientList[Series[QPochhammer[-I*x, x], {x, 0, 100}], x]] (* Vaclav Kotesovec, Sep 09 2017 *)

Formula

(-i*x; x)_inf is the g.f. for A292042(n) + i*a(n).
a(n) = -A292043(n).

A292138 G.f.: Im(1/(-i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

Original entry on oeis.org

0, -1, -1, 0, 0, 0, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 1, -1, -2, -2, -4, -6, -7, -8, -10, -13, -14, -14, -15, -17, -17, -15, -15, -16, -14, -10, -8, -6, -1, 5, 10, 14, 21, 31, 38, 43, 53, 64, 71, 77, 86, 97, 104, 108, 115, 124, 127, 125, 127, 130, 125, 116, 110, 103, 89
Offset: 0

Views

Author

Seiichi Manyama, Sep 09 2017

Keywords

Examples

			Product_{k>=1} 1/(1 + i*x^k) = 1 + (0-1i)*x + (-1-1i)*x^2 + (-1+0i)*x^3 + (-1+0i)*x^4 + (-1+0i)*x^5 + (-2+1i)*x^6 + (-1+2i)*x^7 + ...

Crossrefs

Cf. A292042, A292052, A292136, A292137.

Formula

1/(-i*x; x)_inf is the g.f. for A292136(n) + i*a(n).
a(n) = -A292137(n).
Showing 1-8 of 8 results.

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