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 1-9 of 9 results.

A346894 Expansion of e.g.f. 1 / (1 - (exp(x) - 1)^3 / 3!).

Original entry on oeis.org

1, 0, 0, 1, 6, 25, 110, 721, 6286, 57625, 541470, 5558641, 64351166, 819480025, 11140978030, 160711583761, 2472834185646, 40597082635225, 706816137889790, 12974021811748081, 250395124862965726, 5074637684604691225, 107798916619788396750
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 06 2021

Keywords

Links

Crossrefs

Cf. A000670, A330047, A346895, A346920.
Cf. A000392, A327504, A346922, A353774.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[1/(1 - (Exp[x] - 1)^3/3!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] StirlingS2[k, 3] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(1/(1-(exp(x)-1)^3/3!))) \\ Michel Marcus, Aug 06 2021
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (3*k)!*x^(3*k)/(6^k*prod(j=1, 3*k, 1-j*x)))) \\ Seiichi Manyama, May 07 2022
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, binomial(i, j)*stirling(j, 3, 2)*v[i-j+1])); v; \\ Seiichi Manyama, May 07 2022
  • PARI
    a(n) = sum(k=0, n\3, (3*k)!*stirling(n, 3*k, 2)/6^k); \\ Seiichi Manyama, May 07 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * Stirling2(k,3) * a(n-k).
a(n) ~ n! / (3*(1 + 6^(-1/3)) * log(1 + 6^(1/3))^(n+1)). - Vaclav Kotesovec, Aug 08 2021
From Seiichi Manyama, May 07 2022: (Start)
G.f.: Sum_{k>=0} (3*k)! * x^(3*k)/(6^k * Product_{j=1..3*k} (1 - j * x)).
a(n) = Sum_{k=0..floor(n/3)} (3*k)! * Stirling2(n,3*k)/6^k. (End)

A346921 Expansion of e.g.f. 1 / (1 - log(1 - x)^2 / 2).

Original entry on oeis.org

1, 0, 1, 3, 17, 110, 874, 8064, 85182, 1012248, 13369026, 194245590, 3079135806, 52880064588, 978038495316, 19381794788160, 409702099828104, 9201877089355584, 218832476773294008, 5493266481129425064, 145153549897858762776, 4027310838211114515600
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 07 2021

Keywords

Links

Crossrefs

Cf. A007840, A346922, A346923, A346924.
Cf. A000254, A052811, A305306, A330047, A347001.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[1/(1 - Log[1 - x]^2/2), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Abs[StirlingS1[k, 2]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(1/(1-log(1-x)^2/2))) \\ Michel Marcus, Aug 07 2021
  • PARI
    a(n) = sum(k=0, n\2, (2*k)!*abs(stirling(n, 2*k, 1))/2^k); \\ Seiichi Manyama, May 06 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * |Stirling1(k,2)| * a(n-k).
a(n) ~ n! * exp(sqrt(2)*n) / (sqrt(2) * (exp(sqrt(2)) - 1)^(n+1)). - Vaclav Kotesovec, Aug 08 2021
a(n) = Sum_{k=0..floor(n/2)} (2*k)! * |Stirling1(n,2*k)|/2^k. - Seiichi Manyama, May 06 2022

A347002 Expansion of e.g.f. exp( -log(1 - x)^3 / 3! ).

Original entry on oeis.org

1, 0, 0, 1, 6, 35, 235, 1834, 16352, 163764, 1818030, 22143726, 293476326, 4203311892, 64682865156, 1064154324024, 18636296872320, 346103784493560, 6793394350116600, 140508244952179200, 3054120126193160280, 69596730438090806880, 1659041650323705102840
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 10 2021

Keywords

Links

Crossrefs

Cf. A000399, A327504, A346922, A347001, A347003, A347004.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Exp[-Log[1 - x]^3/3!], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] Abs[StirlingS1[k, 3]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]
  • PARI
    a(n) = sum(k=0, n\3, (3*k)!*abs(stirling(n, 3*k, 1))/(6^k*k!)); \\ Seiichi Manyama, May 06 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * |Stirling1(k,3)| * a(n-k).
a(n) = Sum_{k=0..floor(n/3)} (3*k)! * |Stirling1(n,3*k)|/(6^k * k!). - Seiichi Manyama, May 06 2022

A346923 Expansion of e.g.f. 1 / (1 - log(1 - x)^4 / 4!).

Original entry on oeis.org

1, 0, 0, 0, 1, 10, 85, 735, 6839, 69804, 784580, 9680000, 130312336, 1901581968, 29895585356, 503657235900, 9051009737834, 172807817059664, 3493189152511608, 74530548004474584, 1673793045085649146, 39467836062718058100, 974939402596817961050, 25177327470510057799550
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 07 2021

Keywords

Links

Crossrefs

Cf. A007840, A346921, A346922, A346924.
Cf. A000454, A346895, A347003, A353119.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[1/(1 - Log[1 - x]^4/4!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Abs[StirlingS1[k, 4]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(1/(1-log(1-x)^4/4!))) \\ Michel Marcus, Aug 07 2021
  • PARI
    a(n) = sum(k=0, n\4, (4*k)!*abs(stirling(n, 4*k, 1))/24^k); \\ Seiichi Manyama, May 06 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * |Stirling1(k,4)| * a(n-k).
a(n) ~ n! * 2^(-5/4) * 3^(1/4) / (exp(2^(3/4)*3^(1/4)) * (1 - exp(-2^(3/4)*3^(1/4)))^(n+1)). - Vaclav Kotesovec, Aug 08 2021
a(n) = Sum_{k=0..floor(n/4)} (4*k)! * |Stirling1(n,4*k)|/24^k. - Seiichi Manyama, May 06 2022

A346924 Expansion of e.g.f. 1 / (1 + log(1 - x)^5 / 5!).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 15, 175, 1960, 22449, 269577, 3430790, 46480830, 671260876, 10329270952, 169125055736, 2940784282800, 54182845939104, 1055291277366108, 21674715826211532, 468366193441002564, 10624074081842024496, 252432685158931968768, 6270222495850552958004
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 07 2021

Keywords

Links

Crossrefs

Cf. A007840, A346921, A346922, A346923.
Cf. A000482, A347004, A353200.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[1/(1 + Log[1 - x]^5/5!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Abs[StirlingS1[k, 5]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(1/(1+log(1-x)^5/5!))) \\ Michel Marcus, Aug 07 2021
  • PARI
    a(n) = sum(k=0, n\5, (5*k)!*abs(stirling(n, 5*k, 1))/120^k); \\ Seiichi Manyama, May 06 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * |Stirling1(k,5)| * a(n-k).
a(n) ~ n! * 2^(3/5) * 3^(1/5) * exp(2^(3/5)*15^(1/5)*n) / (5^(4/5) * (exp(2^(3/5)*15^(1/5)) - 1)^(n+1)). - Vaclav Kotesovec, Aug 08 2021
a(n) = Sum_{k=0..floor(n/5)} (5*k)! * |Stirling1(n,5*k)|/120^k. - Seiichi Manyama, May 06 2022

A353118 Expansion of e.g.f. 1/(1 + log(1 - x)^3).

Original entry on oeis.org

1, 0, 0, 6, 36, 210, 2070, 24864, 310632, 4337544, 68922360, 1205002656, 22844264256, 469287123552, 10397824478496, 246800350393344, 6246190572981120, 167972669001740160, 4783274802508890240, 143775432034543203840, 4548946867429143444480
Offset: 0

Views

Author

Seiichi Manyama, May 06 2022

Keywords

Links

Crossrefs

Cf. A052811, A353119, A353200.
Cf. A346922.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[1/(1+Log[1-x]^3),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 04 2023 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+log(1-x)^3)))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=6*sum(j=1, i, binomial(i, j)*abs(stirling(j, 3, 1))*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n\3, (3*k)!*abs(stirling(n, 3*k, 1)));

Formula

a(0) = 1; a(n) = 6 * Sum_{k=1..n} binomial(n,k) * |Stirling1(k,3)| * a(n-k).
a(n) = Sum_{k=0..floor(n/3)} (3*k)! * |Stirling1(n,3*k)|.
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / (3 * (exp(1) - 1)^(n+1)). - Vaclav Kotesovec, May 07 2022

A353881 Expansion of e.g.f. 1/(1 + (x * log(1-x))^3 / 36).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 20, 210, 1960, 18900, 194880, 2166780, 26356880, 349806600, 5029088064, 77748751080, 1284349422720, 22551300670080, 419191223208384, 8222848137607680, 169760091173740800, 3679746265902067200, 83564915096633308800, 1984162781781147770880
Offset: 0

Views

Author

Seiichi Manyama, May 09 2022

Keywords

Crossrefs

Cf. A052830, A353880, A353882.
Cf. A346922, A353884.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+(x*log(1-x))^3/36)))
  • PARI
    a(n) = n!*sum(k=0, n\6, (3*k)!*abs(stirling(n-3*k, 3*k, 1))/(36^k*(n-3*k)!));

Formula

a(n) = n! * Sum_{k=0..floor(n/6)} (3*k)! * |Stirling1(n-3*k,3*k)|/(36^k * (n-3*k)!).

A354392 Expansion of e.g.f. 1/(1 + (exp(x) - 1)^3 / 6).

Original entry on oeis.org

1, 0, 0, -1, -6, -25, -70, 119, 4354, 48215, 371610, 1620839, -10665886, -388969945, -6114636710, -65181228841, -325375497726, 5950049261495, 226564100074970, 4447402833379079, 57902620204276834, 258292327155958535, -12701483290229413350
Offset: 0

Views

Author

Seiichi Manyama, May 25 2022

Keywords

Crossrefs

Cf. A354391, A354393, A354394.
Cf. A346894, A346922.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+(exp(x)-1)^3/6)))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-sum(j=1, i, binomial(i, j)*stirling(j, 3, 2)*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n\3, (3*k)!*stirling(n, 3*k, 2)/(-6)^k);

Formula

a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n,k) * Stirling2(k,3) * a(n-k).
a(n) = Sum_{k=0..floor(n/3)} (3*k)! * Stirling2(n,3*k)/(-6)^k.

A354134 Expansion of e.g.f. 1/(1 - log(1 + x)^3/6).

Original entry on oeis.org

1, 0, 0, 1, -6, 35, -205, 1204, -6692, 29084, 17160, -3069924, 61356724, -959574408, 13499619224, -174983776176, 2029529618080, -18417948918640, 36189097244720, 4235753092128480, -157628320980720480, 4166967770825777280, -95152715945973322560
Offset: 0

Views

Author

Seiichi Manyama, May 18 2022

Keywords

Crossrefs

Cf. A006252, A354135.
Cf. A346894, A346922.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-log(1+x)^3/6)))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, binomial(i, j)*stirling(j, 3, 1)*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n\3, (3*k)!*stirling(n, 3*k, 1)/6^k);

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * Stirling1(k,3) * a(n-k).
a(n) = Sum_{k=0..floor(n/3)} (3*k)! * Stirling1(n,3*k)/6^k.
Showing 1-9 of 9 results.

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