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.

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

A330047 Expansion of e.g.f. exp(-x) / (1 - sinh(x)).

Original entry on oeis.org

1, 0, 1, 3, 13, 75, 511, 4053, 36793, 375735, 4262971, 53203953, 724379173, 10684377795, 169713810631, 2888340723453, 52433443111153, 1011340189494255, 20654264750645491, 445249365444296553, 10103533212012216733, 240731286454287293115, 6008902898851584479551
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 28 2019

Keywords

Comments

Inverse binomial transform of A006154.

Links

Crossrefs

Cf. A006154, A009227, A052841, A330046, A346894, A346895, A346921.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Exp[-x]/(1 - Sinh[x]), {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-(exp(x)-1)^2/2))) \\ Seiichi Manyama, May 07 2022
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (2*k)!*x^(2*k)/(2^k*prod(j=1, 2*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, 2, 2)*v[i-j+1])); v; \\ Seiichi Manyama, May 07 2022
  • PARI
    a(n) = sum(k=0, n\2, (2*k)!*stirling(n, 2*k, 2)/2^k); \\ Seiichi Manyama, May 07 2022

Formula

a(n) = Sum_{k=0..n} (-1)^(n - k) * binomial(n,k) * A006154(k).
a(n) ~ n! / ((2 + sqrt(2)) * (log(1 + sqrt(2)))^(n+1)). - Vaclav Kotesovec, Dec 03 2019
From Seiichi Manyama, May 07 2022: (Start)
E.g.f.: 1/(1 - (exp(x) - 1)^2 / 2).
G.f.: Sum_{k>=0} (2*k)! * x^(2*k)/(2^k * Product_{j=1..2*k} (1 - j * x)).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * Stirling2(k,2) * a(n-k).
a(n) = Sum_{k=0..floor(n/2)} (2*k)! * Stirling2(n,2*k)/2^k. (End)
a(0) = 1; a(n) = (-1)^n + Sum_{k=1..ceiling(n/2)} binomial(n,2*k-1) * a(n-2*k+1). - Prabha Sivaramannair, Oct 06 2023

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

A347001 Expansion of e.g.f. exp( log(1 - x)^2 / 2 ).

Original entry on oeis.org

1, 0, 1, 3, 14, 80, 544, 4284, 38310, 383256, 4239006, 51345690, 675770028, 9600349824, 146396925648, 2384700728760, 41320373582652, 758780222426592, 14718569154071964, 300706641183038292, 6453691377726073128, 145154958710291611200, 3414131149418742544320
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 10 2021

Keywords

Links

Crossrefs

Cf. A000254, A060311, A305306, A346921, A347002, A347003, A347004.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Exp[Log[1 - x]^2/2], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] Abs[StirlingS1[k, 2]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]
  • PARI
    a(n) = sum(k=0, n\2, (2*k)!*abs(stirling(n, 2*k, 1))/(2^k*k!)); \\ Seiichi Manyama, May 06 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * |Stirling1(k,2)| * a(n-k).
a(n) = Sum_{k=0..floor(n/2)} (2*k)! * |Stirling1(n,2*k)|/(2^k * k!). - Seiichi Manyama, May 06 2022

A052811 A simple grammar: sequences of pairs of cycles.

Original entry on oeis.org

1, 0, 2, 6, 46, 340, 3308, 36288, 460752, 6551424, 103685232, 1803956880, 34247483664, 704301934752, 15598712592864, 370149922235520, 9369093828260736, 251968378971718656, 7174943434198029312
Offset: 0

Views

Author

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

Keywords

Comments

Stirling transform of (-1)^n*a(n)=[0,2,-6,46,-340,...] is A005359(n)=[0,2,0,24,0,...]. - Michael Somos, Mar 04 2004

Links

Crossrefs

Cf. A346921.

Programs

  • Maple
    spec := [S,{B=Cycle(Z),C=Prod(B,B),S=Sequence(C)},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((-1)^n*sum(stirling1(n, 2*k)*(2*k)!,k,0,floor(n/2)), n, 0, 18); /* Bruno Berselli, May 25 2011 */
  • PARI
    a(n)=if(n<0,0,n!*polcoeff(1/(1-log(1-x)^2),n))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=2*sum(j=1, i, binomial(i, j)*abs(stirling(j, 2, 1))*v[i-j+1])); v; \\ Seiichi Manyama, May 06 2022
  • PARI
    a(n) = sum(k=0, n\2, (2*k)!*abs(stirling(n, 2*k, 1))); \\ Seiichi Manyama, May 06 2022

Formula

a(n) = (-1)^n*Sum_{k=0..floor(n/2)} Stirling1(n, 2*k)*(2*k)!. - Vladeta Jovovic, Sep 22 2003
E.g.f.: 1/(1-log(1-x)^2).
a(n) = D^n(1/(1-x^2)) evaluated at x = 0, where D is the operator exp(x)*d/dx. Cf. A006252. - Peter Bala, Nov 25 2011
a(n) ~ n!/2 * exp(n)/(exp(1)-1)^(n+1). - Vaclav Kotesovec, Sep 30 2013
a(0) = 1; a(n) = 2 * Sum_{k=1..n} binomial(n,k) * |Stirling1(k,2)| * a(n-k). - Seiichi Manyama, May 06 2022

A353880 Expansion of e.g.f. 1/(1 - (x * log(1-x))^2 / 4).

Original entry on oeis.org

1, 0, 0, 0, 6, 30, 165, 1050, 10192, 108864, 1230660, 14758920, 195861996, 2852815680, 44880446520, 753211040400, 13458760362720, 255688784416800, 5149255813778160, 109489194918180000, 2450182706364430080, 57567025900160259840, 1417073899136197468320
Offset: 0

Views

Author

Seiichi Manyama, May 09 2022

Keywords

Crossrefs

Cf. A052830, A353881, A353882.
Cf. A346921, A353883.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-(x*log(1-x))^2/4)))
  • PARI
    a(n) = n!*sum(k=0, n\4, (2*k)!*abs(stirling(n-2*k, 2*k, 1))/(4^k*(n-2*k)!));

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (2*k)! * |Stirling1(n-2*k,2*k)|/(4^k * (n-2*k)!).

A346966 Expansion of e.g.f. -log( 1 - log(1 - x)^2 / 2 ).

Original entry on oeis.org

1, 3, 14, 80, 559, 4599, 43665, 470196, 5666586, 75600690, 1106587008, 17636532264, 304092954138, 5640892517610, 112029356591862, 2371963759970352, 53338181764577304, 1269586152655203672, 31891196481381667008, 843109673024218773600, 23400930987874505081160
Offset: 2

Views

Author

Ilya Gutkovskiy, Aug 09 2021

Keywords

Crossrefs

Cf. A000254, A003713, A346921, A346944.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[-Log[1 - Log[1 - x]^2/2], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 2] &
a[n_] := a[n] = Abs[StirlingS1[n, 2]] + (1/n) Sum[Binomial[n, k] Abs[StirlingS1[n - k, 2]] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 2, 22}]

Formula

a(n) = |Stirling1(n,2)| + (1/n) * Sum_{k=1..n-1} binomial(n,k) * |Stirling1(n-k,2)| * k * a(k).
a(n) ~ (n-1)! / (1 - exp(-sqrt(2)))^n. - Vaclav Kotesovec, Jun 04 2022
a(n) = Sum_{k=1..floor(n/2)} (2*k)! * |Stirling1(n,2*k)|/(k * 2^k). - Seiichi Manyama, Jan 23 2025

A354389 Expansion of e.g.f. 1/(1 + log(1 + x)^2 / 2).

Original entry on oeis.org

1, 0, -1, 3, -5, -10, 146, -756, 1086, 23400, -300066, 1855590, 341826, -165915828, 2158958556, -10006622640, -172337345496, 4941605486016, -64365944851512, 339328464492456, 5510899593823176, -157099566384759600, 1059259019507498160, 41957473280879898720
Offset: 0

Views

Author

Seiichi Manyama, May 25 2022

Keywords

Crossrefs

Cf. A346921, A354317, A354390, A354391.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[1/(1+(Log[1+x]^2)/2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 08 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+log(1+x)^2/2)))
  • 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, 2, 1)*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n\2, (2*k)!*stirling(n, 2*k, 1)/(-2)^k);

Formula

a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n,k) * Stirling1(k,2) * a(n-k).
a(n) = Sum_{k=0..floor(n/2)} (2*k)! * Stirling1(n,2*k)/(-2)^k.
Showing 1-9 of 9 results.

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