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 1-10 of 10 results.

A052801 A simple grammar: labeled pairs of sequences of cycles.

Original entry on oeis.org

1, 2, 8, 46, 342, 3108, 33324, 411360, 5741856, 89379120, 1534623936, 28804923024, 586686138384, 12885385945248, 303537419684064, 7633673997722496, 204125888803996800, 5782960189212871680
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Links

Crossrefs

Cf. A215916.
Cf. A007840, A354122, A354123.

Programs

  • Maple
    spec := [S,{C=Cycle(Z),B=Sequence(C),S=Prod(B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
  • Mathematica
    CoefficientList[Series[1/(1+Log[1-x])^2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2013 *)
  • Maxima
    makelist(sum((-1)^(n-k)*stirling1(n, k)*(k+1)!, k, 0, n), n, 0, 17); /* Bruno Berselli, May 25 2011 */

Formula

E.g.f.: 1/(-1+log(-1/(-1+x)))^2.
a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling1(n, k)*(k+1)!. - Vladeta Jovovic, Sep 21 2003
a(n) = D^n(1/(1-x)^2) evaluated at x = 0, where D is the operator exp(x)*d/dx. Cf. A052811. - Peter Bala, Nov 25 2011
a(n) ~ n! * n*exp(n)/(exp(1)-1)^(n+2). - Vaclav Kotesovec, Sep 30 2013
From Anton Zakharov, Aug 07 2016: (Start)
a(n) = A007840(n) + A215916(n).
a(n) = Sum_{k=2..n+1} k!*s(n,k) where s(n,k) is the unsigned Stirling number of the first kind, (A132393). (End)
a(0) = 1; a(n) = Sum_{k=1..n} (k/n + 1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Nov 19 2023

A354120 Expansion of e.g.f. 1/(1 - log(1 + x))^3.

Original entry on oeis.org

1, 3, 9, 30, 114, 492, 2388, 12912, 77016, 503112, 3570552, 27399600, 225729360, 1991996640, 18690559200, 186620451840, 1963991600640, 21914748541440, 255336518292480, 3155705206364160, 40209018105116160, 547746803311864320, 7525926332189130240
Offset: 0

Views

Author

Seiichi Manyama, May 17 2022

Keywords

Comments

a(34) is negative. - Vaclav Kotesovec, Jun 04 2022

Links

Crossrefs

Cf. A006252, A317280, A354121.
Cf. A226515, A354122.

Programs

  • Mathematica
    Table[Sum[(k+2)! * StirlingS1[n,k], {k,0,n}]/2, {n,0,35}] (* Vaclav Kotesovec, Jun 04 2022 *)
With[{nn=30},CoefficientList[Series[1/(1-Log[1+x])^3,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 16 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-log(1+x))^3))
  • PARI
    a(n) = sum(k=0, n, (k+2)!*stirling(n, k, 1))/2;

Formula

a(n) = (1/2) * Sum_{k=0..n} (k + 2)! * Stirling1(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (2 * k/n + 1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Nov 19 2023

A354123 Expansion of e.g.f. 1/(1 + log(1 - x))^4.

Original entry on oeis.org

1, 4, 24, 188, 1804, 20416, 265640, 3901320, 63776280, 1147796160, 22540858080, 479500074720, 10980929163360, 269298981833280, 7040446188020160, 195439047629422080, 5740498087530831360, 177855276360034736640, 5796391124741936993280
Offset: 0

Views

Author

Seiichi Manyama, May 17 2022

Keywords

Crossrefs

Cf. A007840, A052801, A354122.
Cf. A226738, A354121.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+log(1-x))^4))
  • PARI
    a(n) = sum(k=0, n, (k+3)!*abs(stirling(n, k, 1)))/6;

Formula

a(n) = (1/6) * Sum_{k=0..n} (k + 3)! * |Stirling1(n,k)|.
a(n) ~ sqrt(Pi/2) * n^(n + 7/2) / (3 * (exp(1) - 1)^(n+4)). - Vaclav Kotesovec, Jun 04 2022
a(0) = 1; a(n) = Sum_{k=1..n} (3*k/n + 1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Nov 19 2023

A367475 Expansion of e.g.f. 1 / (1 + 2 * log(1 - x))^3.

Original entry on oeis.org

1, 6, 54, 636, 9204, 157584, 3111312, 69533472, 1734229344, 47733263232, 1436801816448, 46942939272960, 1654215709835520, 62533593070755840, 2524077593084160000, 108339176213529384960, 4927173048408858531840, 236673892535088351744000
Offset: 0

Views

Author

Seiichi Manyama, Nov 19 2023

Keywords

Crossrefs

Cf. A088500, A367474.
Cf. A354122, A367471.

Programs

  • Maple
    A367475 := proc(n)
    option remember ;
    if n =0 then
        1;
    else
        2*add((2*k/n + 1) * (k-1)! * binomial(n,k) * procname(n-k),k=1..n) ;
    end if;
end proc:
seq(A367475(n),n=0..70) ; # R. J. Mathar, Dec 04 2023
  • PARI
    a(n) = sum(k=0, n, 2^k*(k+2)!*abs(stirling(n, k, 1)))/2;

Formula

a(n) = (1/2) * Sum_{k=0..n} 2^k * (k+2)! * |Stirling1(n,k)|.
a(0) = 1; a(n) = 2*Sum_{k=1..n} (2*k/n + 1) * (k-1)! * binomial(n,k) * a(n-k).

A376393 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 + log(1-x))^3 ).

Original entry on oeis.org

1, 3, 33, 669, 20130, 808902, 40799514, 2480325810, 176637134184, 14428585258896, 1330156753687152, 136632403748954088, 15476220160149512160, 1916493979349783418192, 257601843144279267685056, 37352685483321694825767120, 5812026059839341212943591168, 965974072760231560672817681280
Offset: 0

Views

Author

Seiichi Manyama, Sep 22 2024

Keywords

Links

Crossrefs

Cf. A367139, A376394.
Cf. A354122.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1+log(1-x))^3)/x))
  • PARI
    a(n) = 3*sum(k=0, n, (3*n+k+2)!*abs(stirling(n, k, 1)))/(3*n+3)!;

Formula

E.g.f. A(x) satisfies A(x) = 1/(1 + log(1 - x*A(x)))^3.
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A367139.
a(n) = (3/(3*n+3)!) * Sum_{k=0..n} (3*n+k+2)! * |Stirling1(n,k)|.

A351739 Expansion of e.g.f. 1/(1 + log(1-x))^x.

Original entry on oeis.org

1, 0, 2, 6, 40, 295, 2688, 28588, 348864, 4802922, 73652110, 1245046836, 23003289912, 461188427544, 9972307487660, 231341792369010, 5731422576446208, 151032969213699536, 4218265874407103640, 124471244064061267032, 3869361472890037713560
Offset: 0

Views

Author

Seiichi Manyama, May 20 2022

Keywords

Crossrefs

Cf. A052801, A052809, A354122, A354123.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+log(1-x))^x))

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A052809(k) * binomial(n-1,k-1) * a(n-k).
a(n) ~ n! * exp(n) / (Gamma(1 - 1/exp(1)) * n^(1/exp(1)) * (exp(1) - 1)^(n + 1 - 1/exp(1))). - Vaclav Kotesovec, Jun 04 2022

A375672 Expansion of e.g.f. 1 / (1 + x * log(1 - x))^3.

Original entry on oeis.org

1, 0, 6, 9, 168, 810, 11592, 103320, 1511808, 19350576, 315908640, 5127930720, 95386497984, 1843728194880, 38978317929600, 866801578406400, 20627303078937600, 516780346452733440, 13695223899883530240, 381043219813390540800, 11135125489382277811200
Offset: 0

Views

Author

Seiichi Manyama, Aug 23 2024

Keywords

Crossrefs

Cf. A052830, A375671.
Cf. A354122, A375661.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+x*log(1-x))^3))
  • PARI
    a(n) = n!*sum(k=0, n\2, (k+2)!*abs(stirling(n-k, k, 1))/(n-k)!)/2;

Formula

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A052830.
a(n) = (n!/2) * Sum_{k=0..floor(n/2)} (k+2)! * |Stirling1(n-k,k)|/(n-k)!.

A375722 Expansion of e.g.f. 1 / (1 + 3 * log(1 - x))^3.

Original entry on oeis.org

1, 9, 117, 1962, 40122, 966276, 26755812, 836862192, 29167596504, 1120629465432, 47044646845848, 2142210019297680, 105154320625284240, 5534780654854980000, 310945503593770489440, 18570787974013838515200, 1174884522886771261079040
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2024

Keywords

Crossrefs

Cf. A354263, A375721.
Cf. A354122, A367475.
Cf. A367473.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+3*log(1-x))^3))
  • PARI
    a(n) = sum(k=0, n, 3^k*(k+2)!*abs(stirling(n, k, 1)))/2;

Formula

a(n) = (1/2) * Sum_{k=0..n} 3^k * (k+2)! * |Stirling1(n,k)|.
a(0) = 1; a(n) = 3 * Sum_{k=1..n} (2*k/n + 1) * (k-1)! * binomial(n,k) * a(n-k).

A375900 E.g.f. satisfies A(x) = 1 / (1 + log(1 - x * A(x)^(1/3)))^3.

Original entry on oeis.org

1, 3, 21, 237, 3738, 76212, 1912350, 57099816, 1979628552, 78224586240, 3472089084072, 171098204829120, 9271248509444544, 548011290335056272, 35095593433694127696, 2421035179995679335360, 178997036386314294247680, 14121215676864610247122560
Offset: 0

Views

Author

Seiichi Manyama, Sep 01 2024

Keywords

Crossrefs

Cf. A052802, A375899.
Cf. A354122.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((serreverse(x*(1+log(1-x)))/x)^3))
  • PARI
    a(n) = 3*sum(k=0, n, (n+k+2)!*abs(stirling(n, k, 1)))/(n+3)!;

Formula

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A052802.
E.g.f.: A(x) = ( (1/x) * Series_Reversion(x * (1 + log(1-x))) )^3.
a(n) = (3/(n+3)!) * Sum_{k=0..n} (n+k+2)! * |Stirling1(n,k)|.

A382830 a(n) = Sum_{k=0..n} binomial(n+k-1,k) * |Stirling1(n,k)| * k!.

Original entry on oeis.org

1, 1, 8, 102, 1804, 40890, 1131108, 36948240, 1391945616, 59411849040, 2833582748160, 149347596487056, 8620256620495584, 540775669746661440, 36636074309252234880, 2665704585421541790720, 207329122282259073044736, 17165075378189396045777280, 1507206260097615729874083840
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 06 2025

Keywords

Crossrefs

Cf. A007840, A052801, A277759, A305919, A354122, A354123.

Programs

  • Mathematica
    Table[Sum[Binomial[n + k - 1, k] Abs[StirlingS1[n, k]] k!, {k, 0, n}], {n, 0, 18}]
Table[n! SeriesCoefficient[1/(1 + Log[1 - x])^n, {x, 0, n}], {n, 0, 18}]

Formula

a(n) = n! * [x^n] 1 / (1 + log(1 - x))^n.
a(n) ~ LambertW(exp(2))^n * n^n / (sqrt(1 + LambertW(exp(2))) * exp(n) * (LambertW(exp(2)) - 1)^(2*n)). - Vaclav Kotesovec, Apr 06 2025
Showing 1-10 of 10 results.

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