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 16 results. Next

A048802 Number of labeled rooted trees of nonempty sets with n points. (Each node is a set of 1 or more points.)

Original entry on oeis.org

1, 3, 16, 133, 1521, 22184, 393681, 8233803, 198342718, 5408091155, 164658043397, 5537255169582, 203840528337291, 8153112960102283, 352079321494938344, 16325961781591781401, 809073412162081974237, 42674870241038732398720, 2386963662244981472850709
Offset: 1

Views

Author

Christian G. Bower, Mar 15 1999

Keywords

Examples

			G.f. = x + 3*x^2 + 16*x^3 + 133*x^4 + 1521*x^5 + 22184*x^6 + 393681*x^7 + ...

Links

Crossrefs

Cf. A036249, A038052, A058863, A052807.

Programs

  • Mathematica
    nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[ ComposeSeries[ Series[t,{x,0,nn}],Series[Exp[x]-1 ,{x,0,nn}]],x]  (* Geoffrey Critzer, Sep 16 2012 *)
  • PARI
    {a(n) = sum( k=1, n, stirling(n, k, 2) * k^(k - 1))}; /* Michael Somos, Jun 09 2012 */
  • PARI
    {a(n) = n! * polcoeff( serreverse( log(1 + x*exp(-x +x*O(x^n))) ),n)}
for(n=1,30,print1(a(n),", ")) \\ Paul D. Hanna, Jan 24 2016

Formula

E.g.f.: B(exp(x)-1) where B is e.g.f. of A000169.
E.g.f.: Series_Reversion( log(1 + x*exp(-x)) ). - Paul D. Hanna, Jan 24 2016
a(n) = Sum_{k=1..n} Stirling2(n, k)*k^(k-1). - Vladeta Jovovic, Sep 17 2003
Stirling transform of A000169. - Michael Somos, Jun 09 2012
a(n) ~ sqrt(1+exp(1)) * n^(n-1) / (exp(n) * (log(1+exp(-1)))^(n-1/2)). - Vaclav Kotesovec, Feb 17 2014

A052813 Expansion of e.g.f.: LambertW(log(1-x))/log(1-x).

Original entry on oeis.org

1, 1, 4, 27, 260, 3280, 51414, 965762, 21175496, 531317520, 15021531840, 472654558992, 16385500397496, 620612495460048, 25500923655523848, 1129909190812470840, 53705490284841870144, 2725878142900911376896
Offset: 0

Views

Author

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

Keywords

Comments

Previous name was: A simple grammar.
Given e.g.f. A(x), log(A(x)) = -log(1-x)*A(x) equals e.g.f. of A052807. - Paul D. Hanna, Jul 19 2006

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 27*x^3/3! + 260*x^4/4! +...
Log(A(x))/A(x) = -log(1-x) = x + 1/2*x^2 + 1/3*x^3 + 1/4*x^4 +...

Links

Crossrefs

Cf. A052807 (log(A(x))).

Programs

  • Maple
    spec := [S,{C=Cycle(Z),S=Set(B),B=Prod(C,S)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
  • Mathematica
    CoefficientList[Series[1/Log[1-x]*LambertW[Log[1-x]], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2013 *)
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1/(1-x+x*O(x^n))^A);n!*polcoeff(A,n)} \\ Paul D. Hanna, Jul 19 2006
  • PARI
    {a(n)=n!*polcoeff(sum(m=0,n,(m+1)^(m-1)/m!*(-log(1-x+x*O(x^n)))^m),n)} \\ Paul D. Hanna, Jun 22 2009
  • PARI
    {a(n) = my(A=1+x); for(i=1,n, A = sum(m=0,n, x^m/m! * prod(k=0,m-1,A + k) +x*O(x^n)) ); n!*polcoeff(A,n)}
for(n=0,20, print1(a(n),", ")) \\ Paul D. Hanna, Oct 26 2015
  • PARI
    x='x+O('x^30); Vec(serlaplace((1/log(1-x))*lambertw(log(1-x)))) \\ G. C. Greubel, Feb 19 2018

Formula

a(n) = Sum_{k=0..n} |Stirling1(n, k)|*(k+1)^(k-1). - Vladeta Jovovic, Nov 12 2003
E.g.f. A(x) satisfies: A(x) = 1/(1-x)^A(x). - Paul D. Hanna, Jul 19 2006
E.g.f.: Sum_{n>=0} (n+1)^(n-1)*(-log(1-x))^n/n!. - Paul D. Hanna, Jun 22 2009
E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * Sum_{k=0..n} |Stirling1(n, k)|*A(x)^k. - Paul D. Hanna, Jan 16 2013
a(n) ~ n^(n-1) * exp(3/2+n*exp(-1)-n) / (exp(exp(-1))-1)^(n-1/2). - Vaclav Kotesovec, Sep 30 2013
E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * Product_{k=0..n-1} (A(x) + k). - Paul D. Hanna, Oct 26 2015

Extensions

New name using e.g.f. from Vaclav Kotesovec, Sep 30 2013

A305819 Expansion of e.g.f. 1/(1 + LambertW(-log(1 + x))).

Original entry on oeis.org

1, 1, 3, 17, 132, 1334, 16442, 239994, 4041776, 77183328, 1647541632, 38877352392, 1004869488048, 28234217634024, 856830099396840, 27930093941047464, 973269467390922240, 36104568839480990400, 1420556927968241858880, 59088101641333114906944, 2590680379402887359111424
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 18 2018

Keywords

Comments

Inverse Stirling transform of A000312.

Links

Crossrefs

Cf. A000312, A052807, A277489, A282190, A305981.

Programs

  • Maple
    a:=series(1/(1+LambertW(-log(1+x))),x=0,21): seq(n!*coeff(a,x,n),n=0..20); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 20; CoefficientList[Series[1/(1 + LambertW[-Log[1 + x]]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[Sum[StirlingS1[n, k] k^k, {k, n}], {n, 20}]]
  • PARI
    a(n) = sum(k=0, n, k^k*stirling(n, k, 1)); \\ Seiichi Manyama, Feb 05 2022
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(-log(1+x))))) \\ Seiichi Manyama, Feb 05 2022

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k)*k^k.
a(n) ~ n^n / ((exp(exp(-1)) - 1)^(n + 1/2) * exp(n + (1 - exp(-1))/2)). - Vaclav Kotesovec, Aug 18 2018

A120980 E.g.f. satisfies: A(x)^A(x) = 1 + x.

Original entry on oeis.org

1, 1, -2, 9, -68, 740, -10554, 185906, -3891320, 94259952, -2592071760, 79748398752, -2713685928744, 101184283477680, -4102325527316184, 179674073609647080, -8454031849605513024, 425281651659459346944, -22777115050468598701248
Offset: 0

Views

Author

Paul D. Hanna, Jul 20 2006

Keywords

Links

Crossrefs

Cf. A008275, A052813, A052807, A349561, A349583, A349585.

Programs

  • Mathematica
    CoefficientList[Series[Log[1+x]/LambertW[Log[1+x]], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 09 2013 *)
Table[StirlingS1[n, 0] + StirlingS1[n, 1] + Sum[(-1)^(k + 1)*StirlingS1[n, k]*(k - 1)^(k - 1), {k, 2, n}], {n,0,50}] (* G. C. Greubel, Jun 21 2017 *)
CoefficientList[Series[Exp[LambertW[Log[1+x]]], {x, 0, 25}], x]* Range[0, 25]! (* G. C. Greubel, Jun 22 2017 *)
  • PARI
    {a(n)=local(A=[1,1]);for(i=1,n,A=concat(A,0); A[ #A]=-Vec(Ser(A)^Ser(A))[ #A]);n!*A[n+1]}
  • PARI
    x='x+O('x^50); Vec(serlaplace(exp(lambertw(log(1+x))))) \\ G. C. Greubel, Jun 22 2017

Formula

E.g.f.: A(x) = log(1+x)/LambertW(log(1+x)).
log(A(x)) = LambertW(log(1+x)).
E.g.f.: A(x) = 1/G(-x) where G(x) = g.f. of A052813.
E.g.f. of A052807 = -log(A(-x)) = -log(1-x)/A(-x).
a(n) = Sum_{k=0..n} (-1)^(k+1)*Stirling1(n,k)*(k-1)^(k-1). - Vladeta Jovovic, Jul 22 2006
|a(n)| ~ exp((exp(-1)-1)*n+3/2) * n^(n-1) / (exp(exp(-1))-1)^(n-1/2). - Vaclav Kotesovec, Jul 09 2013

A277489 Expansion of e.g.f. -LambertW(-log(1+x)).

Original entry on oeis.org

0, 1, 1, 5, 26, 224, 2244, 28496, 417976, 7122384, 136770960, 2937770472, 69626588976, 1806936836184, 50936933449752, 1550292926398680, 50661309325357824, 1769286989373994752, 65762170385201959680, 2591979585702305271552, 107982615297265761991680
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 17 2016

Keywords

Links

Crossrefs

Cf. A001865, A052807, A133297.
Cf. A357337, A357338.

Programs

  • Mathematica
    CoefficientList[Series[-LambertW[-Log[1+x]], {x, 0, 20}], x] * Range[0, 20]!
Table[Sum[StirlingS1[n, k]*k^(k-1), {k, 1, n}], {n, 0, 20}]
  • PARI
    my(x='x+O('x^50)); concat([0], Vec(serlaplace(-lambertw(-log(1+x))))) \\ G. C. Greubel, Jun 21 2017

Formula

a(n) = Sum_{k=1..n} Stirling1(n, k)*k^(k-1).
a(n) ~ (exp(exp(-1))-1)^(1/2-n) * exp(-exp(-1)/2+1/2-n) * n^(n-1).
E.g.f.: Series_Reversion( exp(x * exp(-x)) - 1 ). - Seiichi Manyama, Sep 10 2024

A357265 Expansion of e.g.f. -LambertW(x * log(1-x)).

Original entry on oeis.org

0, 0, 2, 3, 32, 150, 1884, 16380, 249808, 3255336, 59596560, 1037413080, 22432698144, 486784686960, 12233449250736, 316660035739320, 9111729094222080, 273147758526888000, 8880267446524694016, 301952732236006556160, 10963551960785051470080
Offset: 0

Views

Author

Seiichi Manyama, Sep 21 2022

Keywords

Links

Crossrefs

Cf. A052807, A355842, A357267.

Programs

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

Formula

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

A305981 Expansion of e.g.f. 1/(1 + LambertW(log(1 - x))).

Original entry on oeis.org

1, 1, 5, 41, 468, 6854, 122582, 2589978, 63129392, 1743732192, 53827681152, 1836453542472, 68620052332752, 2786929842106344, 122241516227220504, 5758920745460806824, 290017142065771138560, 15547326972257789803200, 883974436758296523437760, 53131928820278417749940544, 3366145488853852112016117504
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 18 2018

Keywords

Links

Crossrefs

Cf. A000312, A052807, A277489, A282190, A305819.

Programs

  • Maple
    a:=series(1/(1+LambertW(log(1-x))),x=0,21): seq(n!*coeff(a,x,n),n=0..20); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 20; CoefficientList[Series[1/(1 + LambertW[Log[1 - x]]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[Sum[Abs[StirlingS1[n, k]] k^k, {k, n}], {n, 20}]]
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*k^k*stirling(n, k, 1)); \\ Seiichi Manyama, Feb 05 2022
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(log(1-x))))) \\ Seiichi Manyama, Feb 05 2022

Formula

a(n) = Sum_{k=0..n} |Stirling1(n,k)|*k^k.
a(n) ~ n^n / ((exp(exp(-1)) - 1)^(n + 1/2) * exp(n*(1 - exp(-1)) + 1/2)). - Vaclav Kotesovec, Aug 18 2018

A357334 E.g.f. satisfies A(x) = -log(1 - x) * exp(3 * A(x)).

Original entry on oeis.org

0, 1, 7, 101, 2286, 71064, 2815812, 135719352, 7708432680, 504204903504, 37327594368240, 3085620116373288, 281715917686701264, 28154794766366676888, 3057177180272007475368, 358397769923628731936280, 45115415964514707860498688, 6069465245766845367272738304
Offset: 0

Views

Author

Seiichi Manyama, Sep 24 2022

Keywords

Links

Crossrefs

Cf. A052807, A357333.
Cf. A264408, A357322.

Programs

  • Mathematica
    nmax = 17; A[_] = 0;
Do[A[x_] = -Log[1 - x]*Exp[3*A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 05 2024 *)
  • PARI
    my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(3*log(1-x))/3)))
  • PARI
    a(n) = sum(k=1, n, (3*k)^(k-1)*abs(stirling(n, k, 1)));

Formula

E.g.f.: -LambertW(3 * log(1 - x))/3.
a(n) = Sum_{k=1..n} (3 * k)^(k-1) * |Stirling1(n,k)|.
a(n) ~ n^(n-1) / (sqrt(3) * (exp(exp(-1)/3) - 1)^(n - 1/2) * exp(n - n*exp(-1)/3 - 1/2)). - Vaclav Kotesovec, Oct 04 2022
E.g.f.: Series_Reversion( 1 - exp(-x * exp(-3*x)) ). - Seiichi Manyama, Sep 10 2024

A357333 E.g.f. satisfies A(x) = -log(1 - x) * exp(2 * A(x)).

Original entry on oeis.org

0, 1, 5, 50, 778, 16604, 451668, 14947568, 582982160, 26187136128, 1331445995520, 75589772147328, 4739901861071232, 325353447339098112, 24264683011603485696, 1953776475810372817920, 168924939633683095452672, 15609228287753846217412608
Offset: 0

Views

Author

Seiichi Manyama, Sep 24 2022

Keywords

Links

Crossrefs

Cf. A052807, A357334.
Cf. A264407, A357321.

Programs

  • Mathematica
    nmax = 17; A[_] = 0;
Do[A[x_] = -Log[1 - x]*Exp[2*A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 05 2024 *)
  • PARI
    my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(2*log(1-x))/2)))
  • PARI
    a(n) = sum(k=1, n, (2*k)^(k-1)*abs(stirling(n, k, 1)));

Formula

E.g.f.: -LambertW(2 * log(1 - x))/2.
a(n) = Sum_{k=1..n} (2 * k)^(k-1) * |Stirling1(n,k)|.
a(n) ~ n^(n-1) / (sqrt(2) * (exp(exp(-1)/2)-1)^(n - 1/2) * exp(n - n*exp(-1)/2 - 1/2)). - Vaclav Kotesovec, Oct 04 2022
E.g.f.: Series_Reversion( 1 - exp(-x * exp(-2*x)) ). - Seiichi Manyama, Sep 09 2024

A357343 E.g.f. satisfies A(x) = -log(1 - x * exp(A(x))) * exp(A(x)).

Original entry on oeis.org

0, 1, 5, 53, 878, 19904, 573984, 20112770, 829953368, 39425517072, 2119169565120, 127163052628512, 8426599011632592, 611181716437826832, 48159349246147915944, 4096752391897622411880, 374189567290578072309504, 36525100459236757201316352
Offset: 0

Views

Author

Seiichi Manyama, Sep 25 2022

Keywords

Links

Crossrefs

Cf. A006963, A357344, A357345, A357422.
Cf. A052807, A349556.

Programs

  • PARI
    a(n) = sum(k=1, n, (n+k)^(k-1)*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=1..n} (n+k)^(k-1) * |Stirling1(n,k)|.
E.g.f.: Series_Reversion( exp(-x) * (1 - exp(-x * exp(-x))) ). - Seiichi Manyama, Sep 09 2024
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