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.

A363022 Expansion of Sum_{k>0} x^(2*k)/(1+x^k)^3.

Original entry on oeis.org

0, 1, -3, 7, -10, 13, -21, 35, -39, 36, -55, 85, -78, 71, -118, 155, -136, 130, -171, 232, -234, 177, -253, 389, -310, 248, -390, 455, -406, 378, -465, 651, -586, 426, -626, 832, -666, 533, -822, 1040, -820, 734, -903, 1129, -1144, 783, -1081, 1637, -1197, 961, -1414, 1580, -1378
Offset: 1

Views

Author

Seiichi Manyama, Jun 11 2023

Keywords

Links

Crossrefs

Cf. A325940, A363598, A363613, A363614.
Cf. A002129, A069153, A321543.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (-1)^# * Binomial[#, 2] &]; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)
  • PARI
    my(N=60, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1+x^k)^3)))
  • PARI
    a(n) = sumdiv(n, d, (-1)^d*binomial(d, 2));

Formula

G.f.: Sum_{k>0} binomial(k,2) * (-x)^k/(1 - x^k).
a(n) = Sum_{d|n} (-1)^d * binomial(d,2) = (A002129(n) - A321543(n))/2.

A128315 Inverse Moebius transform of signed A007318.

Original entry on oeis.org

1, 0, 1, 2, -2, 1, -1, 4, -3, 1, 2, -4, 6, -4, 1, 0, 4, -9, 10, -5, 1, 2, -6, 15, -20, 15, -6, 1, -2, 11, -24, 36, -35, 21, -7, 1, 3, -10, 29, -56, 70, -56, 28, -8, 1, 0, 6, -30, 80, -125, 126, -84, 36, -9, 1, 2, -10, 45, -120, 210, -252, 210, -120, 45, -10, 1, -2, 18, -67, 176, -335, 463, -462, 330, -165, 55, -11, 1
Offset: 1

Views

Author

Gary W. Adamson, Feb 25 2007

Keywords

Comments

A128316 = A000012 * A128315.

Examples

			First few rows of the triangle:
   1;
   0,  1;
   2, -2,  1;
  -1,  4, -3,  1;
   2, -4,  6, -4,  1;
   0,  4, -9, 10, -5, 1;
  ...

Links

Crossrefs

Cf. A000034, A000984, A007318, A010693, A048272, A051731, A128316.
Cf. A130595, A325940, A363615, A363616.
Cf. A000012 (row sums).

Programs

  • Magma
    A128315:= func< n,k | (&+[0^(n mod j)*(-1)^(k+j)*Binomial(j-1, k-1): j in [k..n]]) >;
[A128315(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jun 22 2024
  • Mathematica
    A128315[n_, k_]:= (-1)^k*DivisorSum[n, (-1)^#*Binomial[#-1, k-1] &];
Table[A128315[n,k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Jun 22 2024 *)
  • SageMath
    def A128315(n,k): return sum( 0^(n%j)*(-1)^(k+j)*binomial(j-1,k-1) for j in range(k,n+1))
flatten([[A128315(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Jun 22 2024

Formula

T(n, k) = A051731(n, k) * A130595(n-1, k-1) as infinite lower triangular matrices.
T(n, 1) = A048272(n).
Sum_{k=1..n} T(n, k) = A000012(n) = 1 (row sums).
From G. C. Greubel, Jun 22 2024: (Start)
T(n, k) = (-1)^k * Sum_{d|n} (-1)^d * binomial(d-1, k-1).
T(n, 2) = A325940(n), n >= 2.
T(n, 3) = A363615(n), n >= 3.
T(n, 4) = A363616(n), n >= 4.
T(2*n-1, n) = (-1)^(n-1)*A000984(n-1), n >= 1.
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^(n-1)*A081295(n).
Sum_{k=1..n} k*T(n, k) = A000034(n-1), n >= 1.
Sum_{k=1..n} (k+1)*T(n, k) = A010693(n-1), n >= 1. (End)

Extensions

a(43) = 28 inserted and more terms from Georg Fischer, Jun 05 2023

A363598 Expansion of Sum_{k>0} x^(2*k)/(1+x^k)^4.

Original entry on oeis.org

0, 1, -4, 11, -20, 32, -56, 95, -124, 146, -220, 328, -364, 400, -584, 775, -816, 881, -1140, 1486, -1600, 1552, -2024, 2712, -2620, 2562, -3400, 4064, -4060, 4112, -4960, 6231, -6208, 5730, -7216, 8947, -8436, 8000, -10248, 12230, -11480, 11232, -13244, 15752
Offset: 1

Views

Author

Seiichi Manyama, Jun 11 2023

Keywords

Links

Crossrefs

Cf. A325940, A363022, A363613, A363614.
Cf. A002129, A138503, A363604.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (-1)^# * Binomial[# + 1, 3] &]; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)
  • PARI
    my(N=50, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1+x^k)^4)))
  • PARI
    a(n) = sumdiv(n, d, (-1)^d*binomial(d+1, 3));

Formula

G.f.: Sum_{k>0} binomial(k+1,3) * (-x)^k/(1 - x^k).
a(n) = Sum_{d|n} (-1)^d * binomial(d+1,3) = (A002129(n) - A138503(n))/6.

A363615 Expansion of Sum_{k>0} x^(3*k)/(1+x^k)^3.

Original entry on oeis.org

0, 0, 1, -3, 6, -9, 15, -24, 29, -30, 45, -67, 66, -63, 98, -129, 120, -117, 153, -204, 206, -165, 231, -341, 282, -234, 354, -417, 378, -354, 435, -594, 542, -408, 582, -770, 630, -513, 770, -966, 780, -702, 861, -1071, 1072, -759, 1035, -1527, 1143, -930, 1346
Offset: 1

Views

Author

Seiichi Manyama, Jun 11 2023

Keywords

Links

Crossrefs

Cf. A002129, A048272, A128315, A325940, A321543, A363610, A363616.

Programs

  • Magma
    A363615:= func< n | -(&+[(-1)^d*Binomial(d-1,2): d in Divisors(n)]) >;
[A363615(n): n in [1..60]]; // G. C. Greubel, Jun 22 2024
  • Mathematica
    a[n_] := -DivisorSum[n, (-1)^#*Binomial[# - 1, 2] &]; Array[a, 50] (* Amiram Eldar, Jul 18 2023 *)
  • PARI
    my(N=60, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, x^(3*k)/(1+x^k)^3)))
  • PARI
    a(n) = -sumdiv(n, d, (-1)^d*binomial(d-1, 2));
  • SageMath
    def A363615(n): return sum(0^(n%j)*(-1)^(j+1)*binomial(j-1,2) for j in range(1, n+1))
[A363615(n) for n in range(1,61)] # G. C. Greubel, Jun 22 2024

Formula

G.f.: -Sum_{k>0} binomial(k-1,2) * (-x)^k/(1 - x^k).
a(n) = -Sum_{d|n} (-1)^d * binomial(d-1,2).
a(n) = A128315(n, 3), for n >= 3. - G. C. Greubel, Jun 22 2024
a(n) = (A321543(n) - 3*A002129(n) + 2*A048272(n)) / 2. - Amiram Eldar, Jan 04 2025

A363616 Expansion of Sum_{k>0} x^(4*k)/(1+x^k)^4.

Original entry on oeis.org

0, 0, 0, 1, -4, 10, -20, 36, -56, 80, -120, 176, -220, 266, -368, 491, -560, 634, -816, 1050, -1160, 1210, -1540, 1982, -2028, 2080, -2656, 3192, -3276, 3380, -4060, 4986, -5080, 4896, -6008, 7345, -7140, 6954, -8656, 10224, -9880, 9796, -11480, 13552, -13668, 12650
Offset: 1

Views

Author

Seiichi Manyama, Jun 11 2023

Keywords

Links

Crossrefs

Cf. A002129, A128315, A138503, A321543, A325940, A363611, A363615.

Programs

  • Magma
    A363616:= func< n | (&+[(-1)^d*Binomial(d-1,3): d in Divisors(n)]) >;
[A363616(n): n in [1..60]]; // G. C. Greubel, Jun 22 2024
  • Mathematica
    a[n_] := DivisorSum[n, (-1)^# * Binomial[# - 1, 3] &]; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)
  • PARI
    my(N=50, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(4*k)/(1+x^k)^4)))
  • PARI
    a(n) = sumdiv(n, d, (-1)^d*binomial(d-1, 3));
  • SageMath
    def A363616(n): return sum(0^(n%j)*(-1)^j*binomial(j-1,3) for j in range(4, n+1))
[A363616(n) for n in range(1,61)] # G. C. Greubel, Jun 22 2024

Formula

G.f.: Sum_{k>0} binomial(k-1,3) * (-x)^k/(1 - x^k).
a(n) = Sum_{d|n} (-1)^d * binomial(d-1,3).
a(n) = A128315(n, 4), for n >= 4. - G. C. Greubel, Jun 22 2024
a(n) = -(A138503(n) - 6*A321543(n) + 11*A002129(n) - 6*A048272(n)) / 6. - Amiram Eldar, Jan 04 2025

A363613 Expansion of Sum_{k>0} x^(2*k)/(1+x^k)^5.

Original entry on oeis.org

0, 1, -5, 16, -35, 66, -126, 226, -335, 461, -715, 1082, -1365, 1695, -2420, 3286, -3876, 4581, -5985, 7791, -8986, 9912, -12650, 16242, -17585, 19111, -24086, 29115, -31465, 34106, -40920, 49662, -53080, 55030, -66206, 79412, -82251, 85406, -102640, 119931
Offset: 1

Views

Author

Seiichi Manyama, Jun 11 2023

Keywords

Links

Crossrefs

Cf. A325940, A363022, A363598, A363614.
Cf. A363605.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (-1)^# * Binomial[# + 2, 4] &]; Array[a, 40] (* Amiram Eldar, Jul 25 2023 *)
  • PARI
    my(N=50, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1+x^k)^5)))
  • PARI
    a(n) = sumdiv(n, d, (-1)^d*binomial(d+2, 4));

Formula

G.f.: Sum_{k>0} binomial(k+2,4) * (-x)^k/(1 - x^k).
a(n) = Sum_{d|n} (-1)^d * binomial(d+2,4).

A363614 Expansion of Sum_{k>0} x^(2*k)/(1+x^k)^6.

Original entry on oeis.org

0, 1, -6, 22, -56, 121, -252, 484, -798, 1232, -2002, 3145, -4368, 5937, -8630, 12112, -15504, 19678, -26334, 34902, -42762, 51129, -65780, 84337, -98336, 114388, -143304, 175869, -201376, 230120, -278256, 336744, -379000, 420394, -502250, 598459, -658008, 723065, -855042, 997962, -1086008
Offset: 1

Views

Author

Seiichi Manyama, Jun 11 2023

Keywords

Links

Crossrefs

Cf. A325940, A363022, A363598, A363613.
Cf. A363606.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (-1)^#*Binomial[# + 3, 5] &]; Array[a, 40] (* Amiram Eldar, Jul 18 2023 *)
  • PARI
    my(N=50, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1+x^k)^6)))
  • PARI
    a(n) = sumdiv(n, d, (-1)^d*binomial(d+3, 5));

Formula

G.f.: Sum_{k>0} binomial(k+3,5) * (-x)^k/(1 - x^k).
a(n) = Sum_{d|n} (-1)^d * binomial(d+3,5).

A363629 Expansion of Sum_{k>0} (1/(1+x^k)^2 - 1).

Original entry on oeis.org

-2, 1, -6, 6, -8, 4, -10, 15, -16, 6, -14, 22, -16, 8, -28, 32, -20, 13, -22, 32, -36, 12, -26, 56, -34, 14, -44, 42, -32, 24, -34, 65, -52, 18, -52, 68, -40, 20, -60, 82, -44, 32, -46, 62, -84, 24, -50, 122, -60, 31, -76, 72, -56, 40, -76, 108, -84, 30, -62, 124, -64, 32, -110, 130, -88, 48, -70, 92
Offset: 1

Views

Author

Seiichi Manyama, Jun 12 2023

Keywords

Links

Crossrefs

Cf. A363630, A363631.
Cf. A002129, A048272, A325940.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (-1)^#*(# + 1) &]; Array[a, 100] (* Amiram Eldar, Jul 18 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (-1)^d*(d+1));

Formula

G.f.: Sum_{k>0} (k+1) * (-x)^k/(1 - x^k).
a(n) = Sum_{d|n} (-1)^d * (d+1) = -(A002129(n) + A048272(n)).
Showing 1-8 of 8 results.

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