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 1-6 of 6 results.

A198204 Series reversion of (1 - t*x)*log(1 + x) with respect to x.

Original entry on oeis.org

1, 1, 2, 1, 9, 12, 1, 28, 120, 120, 1, 75, 750, 2100, 1680, 1, 186, 3780, 21840, 45360, 30240, 1, 441, 16856, 176400, 705600, 1164240, 665280, 1, 1016, 69552, 1224720, 8316000, 25280640, 34594560, 17297280, 1, 2295, 272250, 7692300, 82577880, 408648240, 998917920, 1167566400, 518918400
Offset: 1

Views

Author

Peter Bala, Jul 31 2012

Keywords

Comments

This triangle is A133399 read by diagonals.

Examples

			Triangle begins
.n\k.|..0....1.....2......3......4......5
= = = = = = = = = = = = = = = = = = = = =
..1..|..1
..2..|..1....2
..3..|..1....9....12
..4..|..1...28...120....120
..5..|..1...75...750...2100...1680
..6..|..1..186..3780..21840..45360..30240
...

Crossrefs

Cf. A001813, A002691, A052892, A058877, A133386, A133399, A141618.

Programs

  • Mathematica
    Flatten[CoefficientList[CoefficientList[InverseSeries[Series[Log[1 + x]*(1 - t*x),{x,0,9}]], x]*Table[n!, {n,0,9}], t]] (* Peter Luschny, Oct 25 2015 *)

Formula

T(n,k) = k!*binomial(n + k - 1,k)*Stirling2(n,k + 1) (n >= 1, k >=0).
E.g.f.: A(x,t) = series reversion of (1 - t*x)*log(1 + x) w.r.t. x = x + (1 + 2*t)*x^2/2! + (1 + 9*t + 12*t^2)*x^3/3! + ....
Main diagonal A001813, first subdiagonal A002691.
Column 1 A058877, column 2 A133386. Row sums A052892.
1 - t*A(x,t) = x/series reversion of x*(1 - t(exp(x) - 1)) with respect to x. Cf. A141618. - Peter Bala, Oct 22 2015

A371329 E.g.f. satisfies A(x) = (exp(x/(1 - A(x))) - 1)/(1 - A(x)).

Original entry on oeis.org

0, 1, 5, 58, 1099, 28966, 978669, 40349478, 1964141687, 110251617526, 7010830858753, 498111156585670, 39106669556183475, 3362091299430435846, 314139422902048625717, 31696638229827506705254, 3434797595698979061279727, 397852853779288923308578966
Offset: 0

Views

Author

Seiichi Manyama, Mar 19 2024

Keywords

Links

Crossrefs

Cf. A052892, A371330.
Cf. A052886, A376036, A376037.

Programs

  • PARI
    a(n) = sum(k=1, n, (n+2*k-2)!/(n+k-1)!*stirling(n, k, 2));

Formula

a(n) = Sum_{k=1..n} (n+2*k-2)!/(n+k-1)! * Stirling2(n,k).
E.g.f.: Series_Reversion( (1 - x) * log(1 + x * (1 - x)) ). - Seiichi Manyama, Sep 08 2024

A371371 E.g.f. satisfies A(x) = exp(x/(1 - A(x))^2) - 1.

Original entry on oeis.org

0, 1, 5, 61, 1209, 33261, 1171933, 50363293, 2554659761, 149399423101, 9896519640981, 732401926901613, 59890184672573929, 5362586032967290765, 521831581416561627149, 54834132144912233219581, 6188110724712474697469025, 746431260858514472012500701
Offset: 0

Views

Author

Seiichi Manyama, Mar 20 2024

Keywords

Links

Crossrefs

Cf. A368033, A371370.
Cf. A052892.

Programs

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

Formula

E.g.f.: Series_Reversion( (1 - x)^2 * log(1+x) ).
a(n) = Sum_{k=1..n} (2*n+k-2)!/(2*n-1)! * Stirling2(n,k).
a(n) ~ 2^(n-1) * LambertW(exp(1/2))^(2*n-1) * n^(n-1) / (sqrt(LambertW(exp(1/2)) + 1) * exp(n) * (2*LambertW(exp(1/2))-1)^(3*n-1)). - Vaclav Kotesovec, Mar 29 2024

A371342 E.g.f. satisfies A(x) = log(1 + x/(1 - A(x))).

Original entry on oeis.org

0, 1, 1, 5, 38, 404, 5514, 91916, 1810080, 41119704, 1058505600, 30450551592, 968121778128, 33709242522864, 1275738359407680, 52141501470591360, 2288907292892799744, 107405692000948859904, 5365016291068305805440, 284225212617080543066880
Offset: 0

Views

Author

Seiichi Manyama, Mar 19 2024

Keywords

Links

Crossrefs

Cf. A370922, A371326.
Cf. A052842, A052892.

Programs

  • Mathematica
    Table[Sum[(n+k-2)! / (n-1)! * StirlingS1[n,k], {k,1,n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 19 2024 *)
  • PARI
    my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(serreverse((1-x)*(exp(x)-1)))))
  • PARI
    a(n) = sum(k=1, n, (n+k-2)!/(n-1)!*stirling(n, k, 1));

Formula

E.g.f.: Series_Reversion( (1 - x) * (exp(x) - 1) ).
a(n) = Sum_{k=1..n} (n+k-2)!/(n-1)! * Stirling1(n,k).
a(n) ~ LambertW(1)^n * n^(n-1) / (sqrt(1 + LambertW(1)) * exp(n) * (1 - LambertW(1))^(2*n-1)). - Vaclav Kotesovec, Mar 19 2024

A376035 E.g.f. satisfies A(x) = exp(x / (1 - A(x))^3) - 1.

Original entry on oeis.org

0, 1, 7, 118, 3205, 120466, 5790619, 339216046, 23443311049, 1867308836986, 168435092561671, 16971155810393302, 1889194092179682061, 230257485553145337106, 30496977601634473249363, 4361533380688447142658046, 669865656003334085318195089
Offset: 0

Views

Author

Seiichi Manyama, Sep 07 2024

Keywords

Links

Crossrefs

Cf. A274265, A376034, A376036.
Cf. A052892, A371371.

Programs

  • Mathematica
    Table[Sum[(3*n+k-2)!/(3*n-1)! * StirlingS2[n,k], {k,1,n}], {n,0,20}] (* Vaclav Kotesovec, Sep 10 2024 *)
  • PARI
    my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(serreverse((1-x)^3*log(1+x)))))
  • PARI
    a(n) = sum(k=1, n, (3*n+k-2)!/(3*n-1)!*stirling(n, k, 2));

Formula

E.g.f.: Series_Reversion( (1 - x)^3 * log(1+x) ).
a(n) = Sum_{k=1..n} (3*n+k-2)!/(3*n-1)! * Stirling2(n,k).
a(n) ~ 3^(4*n-2) * LambertW(2*exp(1/3)/3)^(3*n-1) * n^(n-1) / (sqrt(1 + LambertW(2*exp(1/3)/3)) * exp(n) * 2^(3*n-1) * (3*LambertW(2*exp(1/3)/3) - 1)^(4*n-1)). - Vaclav Kotesovec, Sep 10 2024

A371330 E.g.f. satisfies A(x) = (exp(x/(1 - A(x))) - 1)/(1 - A(x))^2.

Original entry on oeis.org

0, 1, 7, 112, 2901, 104176, 4788191, 268323756, 17744075761, 1352623086136, 116780496526515, 11263219375425172, 1200239384528276285, 140044340185131990336, 17757626485468691645479, 2431398542489983741458940, 357522675169127219183137737
Offset: 0

Views

Author

Seiichi Manyama, Mar 19 2024

Keywords

Links

Crossrefs

Cf. A052892, A371329.

Programs

  • PARI
    a(n) = sum(k=1, n, (n+3*k-2)!/(n+2*k-1)!*stirling(n, k, 2));

Formula

a(n) = Sum_{k=1..n} (n+3*k-2)!/(n+2*k-1)! * Stirling2(n,k).
E.g.f.: Series_Reversion( (1 - x) * log(1 + x * (1 - x)^2) ). - Seiichi Manyama, Sep 08 2024
Showing 1-6 of 6 results.

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

Content is available under The OEIS End-User License Agreement.