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 1-8 of 8 results.

A346922 Expansion of e.g.f. 1 / (1 + log(1 - x)^3 / 3!).

Original entry on oeis.org

1, 0, 0, 1, 6, 35, 245, 2044, 19572, 210524, 2513760, 33012276, 472963876, 7340889192, 122703087416, 2197496734224, 41979155247520, 852063971170960, 18312093589455440, 415420659953439840, 9920128280950954080, 248735658391768241280, 6533773435848445617600
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 07 2021

Keywords

Links

Crossrefs

Cf. A007840, A346921, A346923, A346924.
Cf. A000399, A346894, A347002, A353118.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[1/(1 + Log[1 - x]^3/3!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Abs[StirlingS1[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+log(1-x)^3/3!))) \\ Michel Marcus, Aug 07 2021
  • PARI
    a(n) = sum(k=0, n\3, (3*k)!*abs(stirling(n, 3*k, 1))/6^k); \\ Seiichi Manyama, May 06 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * |Stirling1(k,3)| * a(n-k).
a(n) ~ n! * 6^(1/3) / (3 * exp(6^(1/3)) * (1 - exp(-6^(1/3)))^(n+1)). - Vaclav Kotesovec, Aug 08 2021
a(n) = Sum_{k=0..floor(n/3)} (3*k)! * |Stirling1(n,3*k)|/6^k. - Seiichi Manyama, May 06 2022

A353200 Expansion of e.g.f. 1/(1 + log(1 - x)^5).

Original entry on oeis.org

1, 0, 0, 0, 0, 120, 1800, 21000, 235200, 2693880, 35947800, 609615600, 12504927600, 281242996320, 6545492073120, 155873050569600, 3849612346944000, 100588974863402880, 2818516832681523840, 84728757269204858880, 2706516690047188416000
Offset: 0

Views

Author

Seiichi Manyama, May 06 2022

Keywords

Links

Crossrefs

Cf. A052811, A353118, A353119.
Cf. A346924.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+log(1-x)^5)))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=120*sum(j=1, i, binomial(i, j)*abs(stirling(j, 5, 1))*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n\5, (5*k)!*abs(stirling(n, 5*k, 1)));

Formula

a(0) = 1; a(n) = 120 * Sum_{k=1..n} binomial(n,k) * |Stirling1(k,5)| * a(n-k).
a(n) = Sum_{k=0..floor(n/5)} (5*k)! * |Stirling1(n,5*k)|.
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / (5 * (exp(1) - 1)^(n+1)). - Vaclav Kotesovec, May 07 2022

A353119 Expansion of e.g.f. 1/(1 - log(1 - x)^4).

Original entry on oeis.org

1, 0, 0, 0, 24, 240, 2040, 17640, 202776, 3066336, 52446720, 933636000, 17416490784, 350580364992, 7719355635264, 184232862777600, 4691944607751936, 126358891201529856, 3591751011211717632, 107772466927523060736, 3408777017097439186944
Offset: 0

Views

Author

Seiichi Manyama, May 06 2022

Keywords

Links

Crossrefs

Cf. A052811, A353118, A353200.
Cf. A346923.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[1/(1-Log[1-x]^4),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 13 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-log(1-x)^4)))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=24*sum(j=1, i, binomial(i, j)*abs(stirling(j, 4, 1))*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n\4, (4*k)!*abs(stirling(n, 4*k, 1)));

Formula

a(0) = 1; a(n) = 24 * Sum_{k=1..n} binomial(n,k) * |Stirling1(k,4)| * a(n-k).
a(n) = Sum_{k=0..floor(n/4)} (4*k)! * |Stirling1(n,4*k)|.
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / (4 * (exp(1) - 1)^(n+1)). - Vaclav Kotesovec, May 07 2022

A353344 Expansion of e.g.f. exp(-log(1 - x)^3).

Original entry on oeis.org

1, 0, 0, 6, 36, 210, 1710, 17304, 194712, 2402184, 32536080, 481094856, 7703580456, 132658888752, 2443228469136, 47904722262144, 995970495769920, 21879712141853760, 506301721998264000, 12306713585213260800, 313441368701926135680, 8345931596469584686080
Offset: 0

Views

Author

Seiichi Manyama, May 06 2022

Keywords

Links

Crossrefs

Column k=3 of A357882.
Cf. A000142, A353358, A353404.
Cf. A347002, A353118, A353664.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-log(1-x)^3)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x)^(-log(1-x)^2)))
  • 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-1, j-1)*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))/k!);

Formula

E.g.f.: (1 - x)^(-(log(1 - x))^2).
a(0) = 1; a(n) = 6 * 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)|/k!.

A353774 Expansion of e.g.f. 1/(1 - (exp(x) - 1)^3).

Original entry on oeis.org

1, 0, 0, 6, 36, 150, 1260, 16926, 197316, 2286150, 32821020, 548528046, 9515702196, 174531124950, 3521913283980, 76969474578366, 1777400236160676, 43405229295464550, 1126972561394470140, 30949983774936839886, 893095888222540548756, 27035433957000465352950
Offset: 0

Views

Author

Seiichi Manyama, May 07 2022

Keywords

Links

Crossrefs

Cf. A000670, A052841, A353775.
Cf. A143815, A346894, A353118, A353664.

Programs

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

Formula

G.f.: Sum_{k>=0} (3*k)! * x^(3*k)/Product_{j=1..3*k} (1 - j * x).
a(0) = 1; a(n) = 6 * 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).
a(n) ~ n! / (6 * log(2)^(n+1)). - Vaclav Kotesovec, May 08 2022

A354229 Expansion of e.g.f. 1/(1 - log(1 + x)^3).

Original entry on oeis.org

1, 0, 0, 6, -36, 210, -630, -5376, 153048, -2194296, 22190760, -93956544, -2677330656, 97821857952, -2019503487456, 27899293618944, -98409183995520, -9548919666829440, 410311098024923520, -10652005874894469120, 176525303194482117120, -46197517147757867520
Offset: 0

Views

Author

Seiichi Manyama, May 20 2022

Keywords

Crossrefs

Cf. A006252, A354230.
Cf. A353118, A354134, A354231.

Programs

  • 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)*stirling(j, 3, 1)*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n\3, (3*k)!*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).

A357881 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (k*j)!* |Stirling1(n,k*j)|.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 0, 2, 14, 0, 1, 0, 0, 6, 88, 0, 1, 0, 0, 6, 46, 694, 0, 1, 0, 0, 0, 36, 340, 6578, 0, 1, 0, 0, 0, 24, 210, 3308, 72792, 0, 1, 0, 0, 0, 0, 240, 2070, 36288, 920904, 0, 1, 0, 0, 0, 0, 120, 2040, 24864, 460752, 13109088, 0, 1, 0, 0, 0, 0, 0, 1800, 17640, 310632, 6551424, 207360912, 0
Offset: 0

Views

Author

Seiichi Manyama, Oct 18 2022

Keywords

Examples

			Square array begins:
  1,   1,   1,   1,   1,   1, ...
  0,   1,   0,   0,   0,   0, ...
  0,   3,   2,   0,   0,   0, ...
  0,  14,   6,   6,   0,   0, ...
  0,  88,  46,  36,  24,   0, ...
  0, 694, 340, 210, 240, 120, ...

Crossrefs

Columns k=0-5 give: A000007, A007840, A052811, A353118, A353119, A353200.
Cf. A357119, A357868, A357882.

Programs

  • PARI
    T(n, k) = sum(j=0, n, (k*j)!*abs(stirling(n, k*j, 1)));
  • PARI
    T(n, k) = if(k==0, 0^n, n!*polcoef(1/(1-(-log(1-x+x*O(x^n)))^k), n));

Formula

For k > 0, e.g.f. of column k: 1/(1 - (-log(1-x))^k).
T(0,k) = 1; T(n,k) = k! * Sum_{j=1..n} binomial(n,j) * |Stirling1(j,k)| * T(n-j,k).

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)|.
Showing 1-8 of 8 results.

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

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