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 1-10 of 11 results. Next

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

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

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)

A292042 G.f.: Re((i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

Original entry on oeis.org

1, 0, 0, -1, -1, -2, -2, -3, -3, -4, -3, -4, -3, -3, -1, -1, 2, 3, 7, 9, 14, 16, 23, 26, 33, 37, 45, 48, 57, 60, 68, 70, 77, 76, 82, 78, 80, 72, 70, 55, 48, 26, 11, -19, -42, -84, -116, -169, -213, -278, -333, -413, -479, -572, -651, -757, -846, -965, -1062
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. A035294, A278399, A278400, A278420, A292043, A292052.

Programs

  • Maple
    N:= 100:
S := convert(series( add( (-1)^n*x^(n*(2*n+1))/(mul(1 - x^k,k = 1..2*n)), n = 0..floor(sqrt(N/2)) ), x, N+1 ), polynom):
seq(coeff(S, x, n), n = 0..N); # Peter Bala, Jan 15 2021
  • Mathematica
    Re[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 a(n) + i*A292043(n).
(-i*x; x)_inf is the g.f. for a(n) + i*A292052(n).
a(n)^2 + A292043(n)^2 = A278420(n). - Vaclav Kotesovec, Sep 08 2017
From Peter Bala, Jan 15 2021: (Start)
G.f.: A(x) = Sum_{n >= 0} (-1)^n*x^(n*(2*n+1))/Product_{k = 1..2*n} (1 - x^k). Cf. A035294.
Conjectural g.f.: A(x) = (1/2)*Sum_{n >= 0} (-x)^(n*(n-1)/2)/Product_{k = 1..n} (1 - x^k). (End)

A278401 G.f.: Re(2/(i; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

Original entry on oeis.org

1, -1, -2, -1, -1, -1, -1, 1, 2, 2, 2, 4, 5, 5, 5, 6, 7, 5, 3, 4, 3, 0, -2, -3, -5, -10, -14, -16, -18, -23, -28, -28, -29, -35, -38, -37, -37, -39, -39, -35, -30, -29, -26, -15, -5, 0, 10, 26, 41, 51, 64, 85, 105, 119, 135, 160, 183, 196, 212, 236, 255, 265
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, A278400, A278402.

Programs

  • Maple
    with(gfun): series( add( (-1)^n*x^(2*n)*(1 - x - x^(2*n+1))/mul(1 - x^k, k = 1..2*n+1), n = 0..50), x, 101): seriestolist(%); # Peter Bala, Feb 08 2021
  • Mathematica
    Re[(2/QPochhammer[I, x] + O[x]^70)[[3]]]

Formula

2/(i; x)_inf is the g.f. for a(n) + i*A278402(n).
G.f.: Sum_{n >= 0} (-1)^n*x^(2*n)*(1 - x - x^(2*n+1))/Product_{k = 1..2*n+1} (1 - x^k). - Peter Bala, Feb 08 2021

A278402 G.f.: Im(2/(i; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

Original entry on oeis.org

1, 1, 0, -1, -1, -1, -3, -3, -2, -2, -2, -2, -1, 1, 1, 2, 5, 7, 7, 8, 11, 12, 12, 13, 15, 16, 14, 12, 12, 11, 6, 2, 1, -3, -10, -17, -21, -27, -37, -45, -50, -57, -68, -77, -81, -86, -96, -102, -101, -103, -108, -109, -103, -97, -95, -88, -71, -54, -42, -24, 5
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, A278400, A278401.

Programs

  • Maple
    with(gfun): series( add( (-1)^n*x^(2*n)*(1 + x - x^(2*n+1))/mul(1 - x^k, k = 1..2*n+1), n = 0..50), x, 101): seriestolist(%); # Peter Bala, Feb 09 2021
  • Mathematica
    Im[(2/QPochhammer[I, x] + O[x]^70)[[3]]]

Formula

2/(i; x)_inf is the g.f. for A278401(n) + i*a(n).
G.f.: Sum_{n >= 0} (-1)^n*x^(2*n)*(1 + x - x^(2*n+1))/Product_{k = 1..2*n+1} (1 - x^k). - Peter Bala, Feb 09 2021

A292049 Triangle read by rows: T(n,k) = (-1)^(k-1) * T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, -1, 0, 1, -1, 0, 1, -2, 0, 1, -2, -1, 0, 1, -3, -1, 0, 1, -3, -2, 0, 1, -4, -3, 0, 1, -4, -4, 1, 0, 1, -5, -5, 1, 0, 1, -5, -7, 2, 0, 1, -6, -8, 3, 0, 1, -6, -10, 5, 0, 1, -7, -12, 6, 1, 0, 1, -7, -14, 9, 1, 0, 1, -8, -16, 11, 2, 0, 1, -8, -19
Offset: 0

Views

Author

Seiichi Manyama, Sep 08 2017

Keywords

Examples

			First few rows are:
  1;
  0, 1;
  0, 1;
  0, 1, -1;
  0, 1, -1;
  0, 1, -2;
  0, 1, -2, -1;
  0, 1, -3, -1;
  0, 1, -3, -2;
  0, 1, -4, -3;
  0, 1, -4, -4, 1.

Links

Crossrefs

Row sums give (-1)*A278400.
Columns 0-1 give A000007, A000012.
Cf. A292047.

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).

A292140 G.f.: Im((2*i; x)_oo), where (a; q)_oo is the q-Pochhammer symbol, i = sqrt(-1).

Original entry on oeis.org

-2, -2, -2, 6, 6, 14, 22, 30, 38, 54, 30, 46, 30, 14, -34, -74, -154, -226, -362, -498, -698, -762, -1058, -1218, -1474, -1634, -1890, -1914, -2074, -2002, -1962, -1570, -1210, -266, 606, 2190, 3454, 6030, 8382, 11926, 15334, 20190, 24758, 30990, 36678, 44134
Offset: 0

Views

Author

Seiichi Manyama, Sep 09 2017

Keywords

Links

Crossrefs

Column k=2 of A292160.
Cf. A278400, A292135, A292141.

Formula

(2*i; x)_oo is the g.f. for A292135(n) + i*a(n).

A292160 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Im((k*i; x)_inf), and (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

Original entry on oeis.org

0, -1, 0, -2, -1, 0, -3, -2, -1, 0, -4, -3, -2, 0, 0, -5, -4, -3, 6, 0, 0, -6, -5, -4, 24, 6, 1, 0, -7, -6, -5, 60, 24, 14, 2, 0, -8, -7, -6, 120, 60, 51, 22, 3, 0, -9, -8, -7, 210, 120, 124, 78, 30, 4, 0, -10, -9, -8, 336, 210, 245, 188, 105, 38, 6, 0, -11, -10
Offset: 0

Views

Author

Seiichi Manyama, Sep 10 2017

Keywords

Examples

			Square array begins:
   0, -1, -2, -3, -4, ...
   0, -1, -2, -3, -4, ...
   0, -1, -2, -3, -4, ...
   0,  0,  6, 24, 60, ...
   0,  0,  6, 24, 60, ...

Links

Crossrefs

Columns k=0..2 give A000004, A278400, A292140.
Rows 0+2 give (-1)*A001477.
Main diagonal gives A292162.
Cf. A292159.
Showing 1-10 of 11 results. Next

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