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 21-30 of 30 results.

A022752 Expansion of 1/Product_{m>=1} (1 - m*q^m)^28.

Original entry on oeis.org

1, 28, 462, 5712, 58289, 516292, 4093670, 29660488, 199276056, 1255092972, 7472840004, 42341686632, 229538522801, 1195827758664, 6009154128786, 29217982425632, 137830326653131, 632273980209340
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 28, g(n) = n. - Seiichi Manyama, Aug 17 2023

Links

Crossrefs

Column k=28 of A297328.
Cf. A078308.

Formula

a(0) = 1; a(n) = (28/n) * Sum_{k=1..n} A078308(k) * a(n-k). - Seiichi Manyama, Aug 17 2023

A022753 Expansion of 1/Product_{m>=1} (1 - m*q^m)^29.

Original entry on oeis.org

1, 29, 493, 6264, 65569, 594906, 4826325, 35745951, 245302938, 1576968409, 9577863060, 55328931365, 305653898806, 1621966962395, 8298721485505, 41068822192297, 197116507655270, 919734407613752
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 29, g(n) = n. - Seiichi Manyama, Aug 17 2023

Links

Crossrefs

Column k=29 of A297328.
Cf. A078308.

Formula

a(0) = 1; a(n) = (29/n) * Sum_{k=1..n} A078308(k) * a(n-k). - Seiichi Manyama, Aug 17 2023

A022754 Expansion of 1/Product_{m>=1} (1 - m*q^m)^30.

Original entry on oeis.org

1, 30, 525, 6850, 73500, 682656, 5663205, 42852150, 300202485, 1968839760, 12192045213, 71771729100, 403849667345, 2181900748410, 11361561151605, 57202802787016, 279230335572240, 1324656422161470
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 30, g(n) = n. - Seiichi Manyama, Aug 17 2023

Links

Crossrefs

Column k=30 of A297328.
Cf. A078308.

Formula

a(0) = 1; a(n) = (30/n) * Sum_{k=1..n} A078308(k) * a(n-k). - Seiichi Manyama, Aug 17 2023

A022755 Expansion of 1/Product_{m>=1} (1 - m*q^m)^31.

Original entry on oeis.org

1, 31, 558, 7471, 82119, 780301, 6615617, 51115125, 365372944, 2443413428, 15419852290, 92459940444, 529685434303, 2912402216693, 15427940560977, 78993195741608, 392010552915543, 1890042591320457
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 31, g(n) = n. - Seiichi Manyama, Aug 17 2023

Links

Crossrefs

Column k=31 of A297328.
Cf. A078308.

Formula

a(0) = 1; a(n) = (31/n) * Sum_{k=1..n} A078308(k) * a(n-k). - Seiichi Manyama, Aug 17 2023

A022756 Expansion of 1/Product_{m>=1} (1 - m*q^m)^32.

Original entry on oeis.org

1, 32, 592, 8128, 91464, 888640, 7695744, 60684736, 442387620, 3015281632, 19383646944, 118336634048, 689923993024, 3859022174784, 20788192441664, 108201765333888, 545685611817866, 2672946940511488
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 32, g(n) = n. - Seiichi Manyama, Aug 16 2023

Links

Crossrefs

Column k=32 of A297328.
Cf. A078308.

Formula

a(0) = 1; a(n) = (32/n) * Sum_{k=1..n} A078308(k) * a(n-k). - Seiichi Manyama, Aug 16 2023

A300786 L.g.f.: log(Product_{k>=1} (1 + k*x^k)) = Sum_{n>=1} a(n)*x^n/n.

Original entry on oeis.org

1, 3, 10, 7, 26, 24, 50, -33, 163, 38, 122, -188, 170, 108, 1580, -1793, 290, -273, 362, -1678, 9404, 3248, 530, -49092, 16251, 14862, 66340, 14000, 842, -135556, 962, -429057, 547172, 258386, 509500, -1392821, 1370, 1043160, 4813052, -8088838, 1682, -9267612, 1850, 8218844, 53396438
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 12 2018

Keywords

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + 10*x^3/3 + 7*x^4/4 + 26*x^5/5 + 24*x^6/6 + 50*x^7/7 - 33*x^8/8 + 163*x^9/9 + 38*x^10/10 + ...
exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 7*x^4 + 15*x^5 + 25*x^6 + 43*x^7 + 64*x^8 + 120*x^9 + 186*x^10 + ... + A022629(n)*x^n + ...

Crossrefs

Cf. A022629, A076717, A078308.

Programs

  • Mathematica
    nmax = 45; Rest[CoefficientList[Series[Log[Product[(1 + k x^k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
nmax = 45; Rest[CoefficientList[Series[Sum[Sum[(-1)^(j + 1) k^j x^(j k)/j, {k, 1, nmax}], {j, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]]
nmax = 45; Rest[CoefficientList[Series[Sum[k^2 x^k/(1 + k x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
a[n_] := Sum[(-d)^(n/d + 1), {d, Divisors[n]}]; Table[a[n], {n, 1, 45}]

Formula

L.g.f.: Sum_{j>=1} Sum_{k>=1} (-1)^(j+1)*k^j*x^(j*k)/j = Sum_{n>=1} a(n)*x^n/n.
G.f.: Sum_{k>=1} k^2*x^k/(1 + k*x^k).
a(n) = Sum_{d|n} (-d)^(n/d+1).

A354340 a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} d^(k/d + 1) )/(k * (n-k)!).

Original entry on oeis.org

1, 7, 38, 264, 1629, 16075, 122366, 1414952, 16076913, 213998983, 2112313774, 53581378400, 664573162941, 9967808211387, 239545427723062, 5933102008956848, 79857813309308609, 2677379355344673255, 44453311791217697686, 1743982053518367438616
Offset: 1

Views

Author

Seiichi Manyama, Aug 15 2022

Keywords

Crossrefs

Cf. A078308, A353992, A354848, A356598.

Programs

  • PARI
    a(n) = n!*sum(k=1, n, sumdiv(k, d, d^(k/d+1))/(k*(n-k)!));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-exp(x)*sum(k=1, N, log(1-k*x^k))))

Formula

a(n) = n! * Sum_{k=1..n} A078308(k)/(k * (n-k)!).
E.g.f.: -exp(x) * Sum_{k>0} log(1-k*x^k).

A358279 a(n) = Sum_{d|n} (d-1)! * d^(n/d).

Original entry on oeis.org

1, 3, 7, 29, 121, 747, 5041, 40433, 362935, 3629433, 39916801, 479006531, 6227020801, 87178326609, 1307674371487, 20922790212353, 355687428096001, 6402373709021811, 121645100408832001, 2432902008212950169, 51090942171709691335, 1124000727778046766849
Offset: 1

Views

Author

Seiichi Manyama, Nov 08 2022

Keywords

Crossrefs

Cf. A038507, A062363, A078308, A217576, A321521, A358280.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (# - 1)! * #^(n/#) &]; Array[a, 22] (* Amiram Eldar, Aug 30 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (d-1)!*d^(n/d));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, k!*x^k/(1-k*x^k)))

Formula

G.f.: Sum_{k>0} k! * x^k/(1 - k * x^k).
If p is prime, a(p) = 1 + p!.

A326121 Expansion of Sum_{k>=1} k^2 * x^(2*k) / (1 - k * x^k).

Original entry on oeis.org

0, 1, 1, 5, 1, 18, 1, 33, 28, 58, 1, 246, 1, 178, 369, 577, 1, 1539, 1, 2774, 2531, 2170, 1, 16706, 3126, 8362, 20413, 35366, 1, 116444, 1, 135425, 178479, 131362, 94933, 1110999, 1, 524650, 1596521, 2530946, 1, 7280892, 1, 8403734, 16364457, 8389138, 1, 78568322, 823544, 43420683
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 10 2019

Keywords

Crossrefs

Cf. A055225, A078308, A087909.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Sum[k^2 x^(2 k)/(1 - k x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (n/#)^# &, # > 1 &], {n, 1, 50}]
  • PARI
    a(n)={sumdiv(n, d, if(d > 1, (n/d)^d))} \\ Andrew Howroyd, Sep 10 2019

Formula

a(n) = Sum_{d|n, d>1} (n/d)^d = Sum_{d|n, d
a(p) = 1, where p is prime.
a(n) = A055225(n) - n.

A356578 Expansion of e.g.f. ( Product_{k>0} 1/(1 - k * x^k) )^x.

Original entry on oeis.org

1, 0, 2, 15, 92, 1050, 8514, 147000, 1546544, 29673000, 478186920, 9011752200, 178483287432, 4205087686800, 91775320005264, 2290742704668600, 63289842765692160, 1696665419122968000, 50287699532618564544, 1549916411848463721600
Offset: 0

Views

Author

Seiichi Manyama, Aug 12 2022

Keywords

Crossrefs

Cf. A078308, A353993, A354848.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, 1-k*x^k)^x))
  • PARI
    a354848(n) = (n-1)!*sumdiv(n, d, d^(n/d+1));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j*a354848(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1, a(1) = 0; a(n) = Sum_{k=2..n} k * A354848(k-1) * binomial(n-1,k-1) * a(n-k).
Previous Showing 21-30 of 30 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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