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.

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A302557 Expansion of 1/(2 - Sum_{k>=0} k!*x^k/(1 + x)^(k+1)).

Original entry on oeis.org

1, 0, 1, 2, 10, 48, 288, 1984, 15660, 139312, 1380484, 15080152, 180017780, 2331038048, 32537274756, 486942025288, 7777172706308, 132025174277392, 2373753512469972, 45059504242538328, 900498975768121972, 18898334957168597184, 415537355533831049572, 9552918187197519923176
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 15 2018

Keywords

Comments

Invert transform of A000166.

Links

Crossrefs

Cf. A000166, A051295, A051296, A259869, A259870, A305275.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[1/(2 - Sum[k! x^k/(1 + x)^(k + 1), {k, 0, nmax}]), {x, 0, nmax}], x]
nmax = 23; CoefficientList[Series[1/(1 - Sum[Round[k!/Exp[1]] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Subfactorial[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]

Formula

G.f.: 1/(1 - Sum_{k>=1} A000166(k)*x^k).
G.f.: 1/(2 - 1/(1 - x^2/(1 - 2*x - 4*x^2/(1 - 4*x - 9*x^2/(1 - 6*x - 16*x^2/(1 - ...)))))), a continued fraction.
a(n) ~ exp(-1) * n! * (1 + 2/n^2 + 6/n^3 + 35/n^4 + 256/n^5 + 2187/n^6 + 21620/n^7 + 243947/n^8 + 3098528/n^9 + 43799819/n^10 + ...), for coefficients see A305275. - Vaclav Kotesovec, Aug 18 2018

A321522 Expansion of Product_{k>=1} (1 + x^k)^((k-1)!).

Original entry on oeis.org

1, 1, 1, 3, 8, 32, 153, 883, 5980, 46660, 411861, 4057263, 44104688, 524243696, 6762188285, 94055795999, 1403061499362, 22342571084082, 378257158227079, 6783952072695685, 128481050502464062, 2562250926987454694, 53668572808754641369, 1177957644341460946099
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 12 2018

Keywords

Links

Crossrefs

Cf. A000142, A051295, A104150, A179327, A261052, A321521.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*binomial((i-1)!, j), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 10 2021
  • Mathematica
    nmax = 23; CoefficientList[Series[Product[(1 + x^k)^((k - 1)!), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d!, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 23}]

Formula

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d! ) * x^k/k).
a(n) ~ (n-1)! * (1 + 1/n + 2/n^2 + 7/n^3 + 34/n^4 + 203/n^5 + 1455/n^6 + 12343/n^7 + 121636/n^8 + 1368647/n^9 + 17343274/n^10 + ...). - Vaclav Kotesovec, Nov 13 2018

A134380 A007318 * A084938.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 8, 5, 1, 1, 24, 19, 7, 1, 1, 89, 71, 34, 9, 1, 1, 415, 290, 150, 53, 11, 1, 1, 2372, 1362, 672, 269, 76, 13, 1, 1, 16072, 7486, 3252, 1319, 436, 103, 15, 1, 1, 125673, 48054, 17618, 6671, 2331, 659, 134, 17, 1
Offset: 0

Views

Author

Gary W. Adamson, Oct 22 2007

Keywords

Comments

Row sums = A134381: (1, 2, 5, 15, 52, 205, 921, ...), binomial transform of A051295.

Examples

			First few rows of the triangle:
  1;
  1,   1;
  1,   3,   1;
  1,   8,   5,   1;
  1,  24,  19,   7,  1;
  1,  89,  71,  34,  9,  1;
  1, 415, 290, 150, 53, 11, 1;
  ...

Crossrefs

Cf. A084938, A051295, A134381.

Formula

Binomial transform of A084938, as infinite lower triangular matrices.

A370379 Number of compositions of n where there are (2*k)!/2 sorts of part k.

Original entry on oeis.org

1, 1, 13, 385, 21061, 1864921, 243833533, 44133789745, 10556951897461, 3223557261840841, 1223184443268467053, 564530822421956927905, 311384269987431969105061, 202282520358685311116600761, 152856358784713560205903602973
Offset: 0

Views

Author

Seiichi Manyama, Feb 17 2024

Keywords

Crossrefs

Cf. A002674, A051295, A370378.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(1/(1-sum(k=1, N, (2*k)!/2*x^k)))

Formula

G.f.: 1 / (1 - Sum_{k>=1} (2*k)!/2 * x^k).
a(0) = 1; a(n) = Sum_{k=1..n} (2*k)!/2 * a(n-k).

A134379 A084938 * A000012.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 5, 3, 1, 15, 15, 9, 4, 1, 54, 54, 30, 14, 5, 1, 235, 235, 115, 51, 20, 6, 1, 1237, 1237, 517, 205, 79, 27, 7, 1, 7790, 7790, 2750, 938, 332, 115, 35, 8, 1, 57581, 57581, 17261, 4973, 1545, 505, 160, 44, 9, 1
Offset: 0

Views

Author

Gary W. Adamson, Oct 22 2007

Keywords

Comments

Left border = A051295.
Row sums = A134378.

Examples

			First few rows of the triangle:
   1;
   1,  1;
   2,  2,  1;
   5,  5,  3,  1;
  15, 15,  9,  4,  1;
  54, 54, 30, 14,  5,  1;
  ...

Crossrefs

Cf. A084938, A134378, A051295.

Formula

A084938 * A000012, partial sum triangle by rows, starting from the right of A084938.

A305535 Expansion of 1/(1 - x/(1 - 2*x/(1 - 2*x/(1 - 4*x/(1 - 4*x/(1 - 6*x/(1 - 6*x/(1 - ...)))))))), a continued fraction.

Original entry on oeis.org

1, 1, 3, 13, 75, 557, 5179, 58589, 784715, 12154061, 213593563, 4195613373, 91031201643, 2160916171181, 55687501548539, 1547866851663261, 46150908197995403, 1469089501918434957, 49722765216242122267, 1782934051704982201469, 67514992620138056010667
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 04 2018

Keywords

Comments

Invert transform of A000165, shifted right one place.

Links

Crossrefs

Cf. A000165, A051295, A051296, A112934, A141307, A292778.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[1/(1 - x/(1 + ContinuedFractionK[-2 Floor[(k + 1)/2] x, 1, {k, 1, nmax}])), {x, 0, nmax}], x]
nmax = 20; CoefficientList[Series[1/(1 - Sum[2^(k - 1) (k - 1)! x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[2^(k - 1) (k - 1)! a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

Formula

a(n) ~ 2^(n-1) * (n-1)!. - Vaclav Kotesovec, Sep 18 2021

A361828 a(0) = 1; a(n+1) = Sum_{k=0..n} k^k * a(n-k).

Original entry on oeis.org

1, 1, 2, 7, 40, 338, 3841, 54821, 939335, 18744832, 426390069, 10881017916, 307686450208, 9546443638409, 322375619648549, 11769010007246745, 461834905502223078, 19384809864763869231, 866564718107731746860, 41102477939620052536314
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2023

Keywords

Crossrefs

Cf. A000312, A051295, A277610.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[1/(1 - x - x*Sum[(k*x)^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 26 2023 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(1/(1-x*(sum(k=0, N, (k*x)^k))))

Formula

G.f.: 1 / (1 - x * Sum_{k>=0} (k*x)^k).
a(n) ~ exp(-1) * n^(n-1). - Vaclav Kotesovec, Mar 26 2023
Previous Showing 11-17 of 17 results.

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