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 1-5 of 5 results.

A128316 Triangle read by rows: A000012 * A128315 as infinite lower triangular matrices.

Original entry on oeis.org

1, 1, 1, 3, -1, 1, 2, 3, -2, 1, 4, -1, 4, -3, 1, 4, 3, -5, 7, -4, 1, 6, -3, 10, -13, 11, -5, 1, 4, 8, -14, 23, -24, 16, -6, 1, 7, -2, 15, -33, 46, -40, 22, -7, 1, 7, 4, -15, 47, -79, 86, -62, 29, -8, 1, 9, -6, 30, -73, 131, -166, 148, -91, 37, -9, 1, 7, 12, -37, 103, -204, 297, -314, 239, -128, 46, -10, 1
Offset: 1

Views

Author

Gary W. Adamson, Feb 25 2007

Keywords

Comments

A128316 * [1,2,3...] = A000034: [1,2,1,2,...].

Examples

			First few rows of the triangle:
  1;
  1,  1;
  3, -1,   1;
  2,  3   -2,   1;
  4, -1,   4,  -3,   1;
  4,  3,  -5,   7,  -4,  1;
  6, -3,  10, -13,  11, -5,  1;
  4,  8, -14,  23, -24, 16, -6, 1;
  ...

Links

Crossrefs

Cf. A000012, A000034, A014300, A026641, A059851, A128315.
Sums include: A000027 (row), A032766, A047215, A344817 (alternating sign).

Programs

  • Magma
    A128316:= func< n,k | (&+[(-1)^(j+k)*Floor(n/j)*Binomial(j-1,k-1): j in [k..n]]) >;
[A128316(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jun 23 2024
  • Mathematica
    T[n_, k_]:= Sum[(-1)^(j+k)*Floor[n/j]*Binomial[j-1,k-1], {j,k,n}];
Table[T[n,k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Jun 23 2024 *)
  • SageMath
    def A128316(n,k): return sum((-1)^(j+k)*int(n//j)*binomial(j-1,k-1) for j in range(k,n+1))
flatten([[A128316(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Jun 23 2024

Formula

Sum_{k=1..n} T(n, k) = A000027(n) (row sums).
T(n, 1) = A059851(n).
From G. C. Greubel, Jun 23 2024: (Start)
T(n, k) = A010766(n,k) * AA130595(n-1, k-1) as infinite lower triangular matrices.
T(n, k) = Sum_{j=k..n} (-1)^(j+k) * floor(n/j) * binomial(j-1, k-1).
T(2*n-1, n) = (-1)^(n-1)*A026641(n).
T(2*n-2, n-1) = (-1)^n*A014300(n-1), for n >= 2.
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = A344817(n).
Sum_{k=1..n} k*T(n, k) = A032766(n-1).
Sum_{k=1..n} (k+1)*T(n, k) = A047215(n). (End)

Extensions

a(28) = 1 inserted and more terms from Georg Fischer, Jun 06 2023

A325940 Expansion of Sum_{k>=1} x^(2*k) / (1 + x^k)^2.

Original entry on oeis.org

0, 1, -2, 4, -4, 4, -6, 11, -10, 6, -10, 18, -12, 8, -20, 26, -16, 13, -18, 28, -28, 12, -22, 48, -28, 14, -36, 38, -28, 24, -30, 57, -44, 18, -44, 62, -36, 20, -52, 74, -40, 32, -42, 58, -72, 24, -46, 110, -54, 31, -68, 68, -52, 40, -68, 100, -76, 30, -58, 116
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 09 2019

Keywords

Links

Crossrefs

Cf. A002129, A048272, A065608, A128315, A363615, A363616.

Programs

  • Magma
    A325940:= func< n | (&+[0^(n mod j)*(-1)^j*(j-1): j in [1..n]]) >;
[A325940(n): n in [1..70]]; // G. C. Greubel, Jun 22 2024
  • Mathematica
    nmax = 60; CoefficientList[Series[Sum[x^(2 k)/(1 + x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[(-1)^d (d - 1), {d, Divisors[n]}], {n, 1, 60}]
  • PARI
    {a(n) = sumdiv(n, d, (-1)^d*(d-1))} \\ Seiichi Manyama, Sep 14 2019
  • SageMath
    def A325940(n): return sum(0^(n%j)*(-1)^j*(j-1) for j in range(1, n+1))
[A325940(n) for n in range(1,71)] # G. C. Greubel, Jun 22 2024

Formula

G.f.: Sum_{k>=2} (-1)^k * (k - 1) * x^k / (1 - x^k).
a(n) = Sum_{d|n} (-1)^d * (d - 1).
a(n) = A048272(n) - A002129(n).
Faster converging series: A(q) = Sum_{n >= 1} (-1)^n*q^(n^2)*((n-1)*q^(3*n) + n*q^(2*n) + (n-2)*q^n + n-1)/((1 + q^n)*(1 - q^(2*n))) - apply the operator t*d/dt to equation 1 in Arndt, then set t = -q and x = q. - Peter Bala, Jan 22 2021
a(n) = A128315(n, 2). - G. C. Greubel, Jun 22 2024

A081295 a(n) = (-1)^(n+1) * coefficient of x^n in Sum_{k>=1} x^k/(1+2*x^k).

Original entry on oeis.org

1, 1, 5, 9, 17, 29, 65, 137, 261, 497, 1025, 2085, 4097, 8129, 16405, 32905, 65537, 130845, 262145, 524793, 1048645, 2096129, 4194305, 8390821, 16777233, 33550337, 67109125, 134225865, 268435457, 536855053, 1073741825, 2147516553, 4294968325, 8589869057
Offset: 1

Views

Author

Benoit Cloitre, Apr 20 2003

Keywords

Links

Crossrefs

Cf. A034729, A128315.

Programs

  • Magma
    A081295:= func< n | (-1)^(n+1)*(&+[(-2)^(d-1): d in Divisors(n)]) >;
[A081295(n): n in [1..40]]; // G. C. Greubel, Jun 22 2024
  • Maple
    f:= n -> (-1)^(n+1)*add((-2)^(d-1),d=numtheory:-divisors(n)):
map(f, [$1..100]); # Robert Israel, Jun 04 2018
  • Mathematica
    A081295[n_]:= (-1)^(n+1)*DivisorSum[n, (-2)^(#-1) &];
Table[A081295[n], {n, 40}] (* G. C. Greubel, Jun 22 2024 *)
  • PARI
    a(n) =if(n<1, 0, (-1)^(n+1)*polcoeff(sum(k=1, n, x^k/(1+2*x^k), x*O(x^n)), n))
  • SageMath
    def A081295(n): return (-1)^(n+1)*sum((-2)^(k-1) for k in (1..n) if (k).divides(n))
[A081295(n) for n in range(1,41)] # G. C. Greubel, Jun 22 2024

Formula

a(n) = (-1)^(n+1) * [x^n]( Sum_{k>=1} x^k/(1+2*x^k) ).
a(p) = 2^(p-1) - 1, for p prime.
a(n) = (-1)^(n+1) * Sum_{d|n} (-2)^(d-1). - Robert Israel, Jun 04 2018
a(n) = (-1)^(n-1)*Sum_{k=1..n} (-1)^(k-1)*A128315(n, k). - G. C. Greubel, Jun 22 2024

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
Showing 1-5 of 5 results.

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