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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A343686 a(0) = 1; a(n) = 3 * n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k).

Original entry on oeis.org

1, 4, 33, 410, 6796, 140824, 3501782, 101589732, 3368237928, 125634319104, 5206805098752, 237370661584704, 11805144854303760, 636030155604374400, 36903603627294958416, 2294156656214759133024, 152126925169297299197184, 10718105879980375520103936, 799564645068022035991527552
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 26 2021

Keywords

Crossrefs

Cf. A007840, A052820, A053486, A343685, A343687.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = 3 n a[n - 1] + Sum[Binomial[n, k] (n - k - 1)! a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[1/(1 - 3 x + Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: 1 / (1 - 3*x + log(1 - x)).
a(n) ~ n! / ((3/c + 2 - c) * (1 - c/3)^n), where c = LambertW(3*exp(2)) = 2.2761339297716461777892556270138... - Vaclav Kotesovec, Apr 26 2021

A343687 a(0) = 1; a(n) = 4 * n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k).

Original entry on oeis.org

1, 5, 51, 782, 15992, 408814, 12541010, 448834728, 18358297416, 844755218400, 43190363326992, 2429044756967520, 149029669269441456, 9905401062535389072, 709016063545908259248, 54375505616232613595904, 4448148376192382963462400, 386619861956492109750650496, 35580548688887294090357622912
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 26 2021

Keywords

Crossrefs

Cf. A007840, A052820, A053487, A343685, A343686.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = 4 n a[n - 1] + Sum[Binomial[n, k] (n - k - 1)! a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[1/(1 - 4 x + Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: 1 / (1 - 4*x + log(1 - x)).
a(n) ~ n! / ((4/c + 3 - c) * (1 - c/4)^n), where c = LambertW(4*exp(3)) = 3.2176447220005493578369738... - Vaclav Kotesovec, Apr 26 2021

A350725 a(n) = Sum_{k=0..n} k! * k^(n-k) * Stirling1(n,k).

Original entry on oeis.org

1, 1, 1, -4, -2, 274, -3442, -12552, 2108664, -63083232, 87416112, 112192496976, -7487840132544, 174521224997040, 19793498724358032, -3109195219736188416, 209306170972547346816, 2973238556525799866496, -3013574861684426837113728, 456220653756733889826621696
Offset: 0

Views

Author

Seiichi Manyama, Feb 03 2022

Keywords

Crossrefs

Cf. A007840, A229234, A320083, A350721, A350726.

Programs

  • Mathematica
    a[0] = 1; a[n_] := Sum[k! * k^(n-k) * StirlingS1[n, k], {k, 1, n}]; Array[a, 20, 0] (* Amiram Eldar, Feb 03 2022 *)
  • PARI
    a(n) = sum(k=0, n, k!*k^(n-k)*stirling(n, k, 1));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, log(1+k*x)^k/k^k)))

Formula

E.g.f.: Sum_{k>=0} log(1 + k*x)^k / k^k.

A352271 Expansion of e.g.f. 1/(2 - exp(x) - log(1 + x)).

Original entry on oeis.org

1, 2, 8, 51, 427, 4485, 56461, 829619, 13929175, 263120293, 5522411441, 127497249825, 3211140897757, 87615489275587, 2574463431688695, 81050546853002151, 2721785052811891411, 97113737702073060713, 3668859532725782696709, 146306156466305491481253
Offset: 0

Views

Author

Seiichi Manyama, Mar 10 2022

Keywords

Crossrefs

Cf. A000670, A007840, A352147, A352270.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(2-exp(x)-log(1+x))))
  • PARI
    a(n) = if(n==0, 1, sum(k=1, n, ((-1)^(k-1)*(k-1)!+1)*binomial(n, k)*a(n-k)));

Formula

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

A353184 Expansion of e.g.f. 1/(1 - Sum_{k>=1} x^(k^2) / (k^2)).

Original entry on oeis.org

1, 1, 2, 6, 30, 180, 1260, 10080, 93240, 1015560, 12146400, 158004000, 2226193200, 34162128000, 565750785600, 10034584560000, 190820565936000, 3845407181616000, 81995523626016000, 1844123531009760000, 43689721287532320000, 1086745683839175360000
Offset: 0

Views

Author

Seiichi Manyama, Apr 29 2022

Keywords

Crossrefs

Cf. A006456, A007840, A353180.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[(k^2 - 1)! * Binomial[n, k^2] * a[n - k^2], {k, 1, Floor@Sqrt[n]}]; Array[a, 22, 0] (* Amiram Eldar, Apr 30 2022 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=1, sqrtint(N), x^k^2/(k^2)))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, sqrtint(i), (j^2-1)!*binomial(i, j^2)*v[i-j^2+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..floor(sqrt(n))} (k^2-1)! * binomial(n,k^2) * a(n-k^2).

A354251 Expansion of e.g.f. Sum_{k>=0} (3*k)! * (-log(1-x))^k / k!.

Original entry on oeis.org

1, 6, 726, 365052, 481186836, 1312477120944, 6422029618230000, 51225621215200895520, 621881012244669445985760, 10911233517605729917096273920, 265743399210784245852461349120000, 8697920910678436598411074217669652480
Offset: 0

Views

Author

Seiichi Manyama, May 21 2022

Keywords

Crossrefs

Cf. A007840, A354244.
Cf. A316748, A353118, A354250.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (3*k)!*(-log(1-x))^k/k!)))
  • PARI
    a(n) = sum(k=0, n, (3*k)!*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=0..n} (3*k)! * |Stirling1(n,k)|.

A355293 Expansion of e.g.f. 1 / (1 - x - x^2/2 - x^3/3).

Original entry on oeis.org

1, 1, 3, 14, 82, 610, 5450, 56700, 674520, 9027480, 134236200, 2195701200, 39180094800, 757389032400, 15767305554000, 351689317980000, 8367381470448000, 211518767796336000, 5661504152255952000, 159954273475764768000, 4757034049019572320000, 148547713504322452320000, 4859583724723970642400000
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 27 2022

Keywords

Crossrefs

Cf. A007840, A057693, A080599, A189886, A355294.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[1/(1 - x - x^2/2 - x^3/3), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = a[1] = 1; a[2] = 3; a[n_] := a[n] = n a[n - 1] + n (n - 1) a[n - 2]/2 + n (n - 1) (n - 2) a[n - 3]/3; Table[a[n], {n, 0, 22}]

Formula

a(n) = n * a(n-1) + n * (n-1) * a(n-2) / 2 + n * (n-1) * (n-2) * a(n-3) / 3.

A355294 Expansion of e.g.f. 1 / (1 - x - x^2/2 - x^3/3 - x^4/4).

Original entry on oeis.org

1, 1, 3, 14, 88, 670, 6170, 66360, 815640, 11272800, 173132400, 2925014400, 53909394000, 1076365290000, 23144112591600, 533193460800000, 13102608591072000, 342105146182800000, 9457689380931792000, 275988880808825184000, 8477631163592791200000, 273430368958004818560000, 9238944655686318693120000
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 27 2022

Keywords

Crossrefs

Cf. A007840, A070945, A080599, A276924, A355293.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[1/(1 - x - x^2/2 - x^3/3 - x^4/4), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = a[1] = 1; a[2] = 3; a[3] = 14; a[n_] := a[n] = n a[n - 1] + n (n - 1) a[n - 2]/2 + n (n - 1) (n - 2) a[n - 3]/3 + n (n - 1) (n - 2) (n - 3) a[n - 4]/4; Table[a[n], {n, 0, 22}]

Formula

a(n) = n * a(n-1) + n * (n-1) * a(n-2) / 2 + n * (n-1) * (n-2) * a(n-3) / 3 + n * (n-1) * (n-2) * (n-3) * a(n-4) / 4.

A355718 Expansion of e.g.f. exp( x/(1 + log(1-x)) ).

Original entry on oeis.org

1, 1, 3, 16, 117, 1071, 11725, 149122, 2158401, 35006941, 628552231, 12372376116, 264849067549, 6124239060915, 152099146415385, 4037206919213686, 114038575520545153, 3415098936831144537, 108065651366801837611, 3602585901321224507992
Offset: 0

Views

Author

Seiichi Manyama, Jul 15 2022

Keywords

Crossrefs

Cf. A355719, A355720.
Cf. A007840, A052860.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/(1+log(1-x)))))
  • PARI
    a007840(n) = sum(k=0, n, k!*abs(stirling(n, k, 1)));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, j*a007840(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A052860(k) * binomial(n-1,k-1) * a(n-k).
a(n) ~ n^(n-1/4) * exp(1/4 - exp(-1) + 2*exp(-1/2)*sqrt(n)) / (sqrt(2) * (exp(1) - 1)^n). - Vaclav Kotesovec, Jul 15 2022

A368283 Expansion of e.g.f. exp(2*x) / (1 + log(1 - x)).

Original entry on oeis.org

1, 3, 11, 52, 320, 2486, 23402, 258252, 3263528, 46433648, 734322672, 12776283136, 242519067056, 4987324250416, 110454579648688, 2621008072506592, 66341399843669760, 1784150447268259456, 50804574646886197888, 1527058892582680257024
Offset: 0

Views

Author

Seiichi Manyama, Dec 19 2023

Keywords

Crossrefs

Cf. A007840, A291979, A330149, A368284.
Cf. A368285.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=2^i+sum(j=1, i, (j-1)!*binomial(i, j)*v[i-j+1])); v;

Formula

a(n) = 2^n + Sum_{k=1..n} (k-1)! * binomial(n,k) * a(n-k).
a(n) ~ n! * exp(n + 2 - 2*exp(-1)) / (exp(1) - 1)^(n+1). - Vaclav Kotesovec, Dec 29 2023
Previous Showing 71-80 of 104 results. Next

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