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.

A052506 Expansion of e.g.f. exp(x*exp(x)-x).

Original entry on oeis.org

1, 0, 2, 3, 16, 65, 336, 1897, 11824, 80145, 586000, 4588001, 38239224, 337611001, 3144297352, 30779387745, 315689119456, 3383159052833, 37790736663456, 439036039824193, 5294386116882280, 66155074120062921, 855156188538926296, 11418964004032623809
Offset: 0

Views

Author

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

Keywords

Comments

a(n) is the number of forests of rooted labeled trees with height exactly one. Equivalently, the number of idempotent mappings from {1,2,...,n} into {1,2,...,n} where each fixed point has at least one (other than itself) element mapped to it. See the second summation formula provided by Vladeta Jovovic with conditions on k, the number of fixed points. - Geoffrey Critzer, Sep 20 2012

Links

Crossrefs

Cf. A000248, A240989.
Cf. A351736, A351737.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x*Exp(x)-x) )); [Factorial(n-1)*b[n]: n in [1..m-1]]; // G. C. Greubel, May 13 2019
  • Maple
    spec := [S,{S=Set(Tree), Tree=Prod(Z,Set(Z,0 < card))},labeled]: seq(combstruct[count](spec, size=n), n=0..20);
  • Mathematica
    nn=20;Range[0,nn]! CoefficientList[Series[Exp[x(Exp[x]-1)], {x,0,nn}], x]  (* Geoffrey Critzer, Sep 20 2012 *)
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace( exp(x*exp(x)-x) )) \\ G. C. Greubel, Nov 15 2017
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-(k-1)*x)^(k+1))) \\ Seiichi Manyama, Aug 29 2022
  • Sage
    m = 30; T = taylor(exp(x*exp(x)-x), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 13 2019

Formula

a(n) = Sum_{k=0..n} binomial(n, k)*(n-k-1)^k. - Vladeta Jovovic, Apr 12 2003
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*k!*Stirling2(n-k, k). - Vladeta Jovovic, Dec 19 2004
a(n) ~ exp((1-r*(n+r))/(1+r)) * n^(n+1/2) * sqrt(1+r) / (r^n * sqrt((1+r)^3 + n*(1+3*r+r^2))), where r satisfies exp(r)*(1+r) - (1+n)/r = 1. - Vaclav Kotesovec, Aug 04 2014
(a(n)/n!)^(1/n) ~ exp(1/(2*LambertW(sqrt(n)/2))) / (2*LambertW(sqrt(n)/2)). - Vaclav Kotesovec, Aug 06 2014
G.f.: Sum_{k>=0} x^k / (1 - (k-1)*x)^(k+1). - Seiichi Manyama, Aug 29 2022

A351737 Expansion of e.g.f. exp( x * (exp(3 * x) - 1) ).

Original entry on oeis.org

1, 0, 6, 27, 216, 2025, 21708, 260253, 3460320, 50395041, 795324420, 13495904829, 244747554912, 4718754452529, 96285948702804, 2071265238290565, 46815054101658432, 1108489016781839169, 27424412680091114628, 707277138662880504045, 18974871706141125008640
Offset: 0

Views

Author

Seiichi Manyama, May 20 2022

Keywords

Links

Crossrefs

Cf. A052506, A351736.
Cf. A351734, A351735.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(exp(3*x)-1))))
  • PARI
    a(n) = n!*sum(k=0, n\2, 3^(n-k)*stirling(n-k, k, 2)/(n-k)!);
  • PARI
    a(n) = sum(k=0, n, (3*k-1)^(n-k)*binomial(n, k)); \\ Seiichi Manyama, Aug 29 2022
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-(3*k-1)*x)^(k+1))) \\ Seiichi Manyama, Aug 29 2022

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} 3^(n-k) * Stirling2(n-k,k)/(n-k)!.
From Seiichi Manyama, Aug 29 2022: (Start)
a(n) = Sum_{k=0..n} (3*k-1)^(n-k) * binomial(n,k).
G.f.: Sum_{k>=0} x^k / (1 - (3*k-1)*x)^(k+1). (End)

A356806 a(n) = Sum_{k=0..n} (k*n-1)^(n-k) * binomial(n,k).

Original entry on oeis.org

1, 0, 4, 27, 448, 10625, 344736, 14437213, 753991680, 47974773393, 3650824000000, 326917384798301, 33956137832546304, 4041303651931462969, 545552768347831566336, 82828479894303251953125, 14040577418634835164921856, 2640293357854435329683551265
Offset: 0

Views

Author

Seiichi Manyama, Aug 29 2022

Keywords

Links

Crossrefs

Cf. A052506, A351736, A351737.
Cf. A356811, A356814, A356817.

Programs

  • PARI
    a(n) = sum(k=0, n, (k*n-1)^(n-k)*binomial(n, k));
  • PARI
    a(n) = n!*sum(k=0, n\2, n^(n-k)*stirling(n-k, k, 2)/(n-k)!);

Formula

a(n) = n! * [x^n] exp( x * (exp(n * x) - 1) ).
a(n) = n! * Sum_{k=0..floor(n/2)} n^(n-k) * Stirling2(n-k,k)/(n-k)!.
a(n) = [x^n] Sum_{k>=0} x^k / (1 - (n*k-1)*x)^(k+1).

A356812 Expansion of e.g.f. exp(x * (1 - exp(2*x))).

Original entry on oeis.org

1, 0, -4, -12, 16, 400, 2208, -448, -131840, -1357056, -4820480, 71120896, 1537308672, 14006460416, 3075702784, -2224350781440, -41354996154368, -359660395495424, 1675436608585728, 121894823709900800, 2317859245604208640, 20543311167964053504
Offset: 0

Views

Author

Seiichi Manyama, Aug 29 2022

Keywords

Links

Crossrefs

Cf. A292893, A356813.
Cf. A351736, A356815, A356819.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[x(1-Exp[2x])],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 04 2023 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(1-exp(2*x)))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (-x)^k/(1-(2*k+1)*x)^(k+1)))
  • PARI
    a(n) = sum(k=0, n, (-1)^k*(2*k+1)^(n-k)*binomial(n, k));
  • PARI
    a(n) = n!*sum(k=0, n\2, (-1)^k*2^(n-k)*stirling(n-k, k, 2)/(n-k)!);

Formula

G.f.: Sum_{k>=0} (-x)^k / (1 - (2*k+1)*x)^(k+1).
a(n) = Sum_{k=0..n} (-1)^k * (2*k+1)^(n-k) * binomial(n,k).
a(n) = n! * Sum_{k=0..floor(n/2)} (-1)^k * 2^(n-k) * Stirling2(n-k,k)/(n-k)!.

A356815 Expansion of e.g.f. exp(-x * (exp(2*x) + 1)).

Original entry on oeis.org

1, -2, 0, 4, 32, 48, -608, -6400, -24064, 163072, 3567104, 28394496, 6535168, -3250745344, -50725740544, -344530853888, 2476610551808, 110057610608640, 1655672654135296, 9616664975114240, -195178079811272704, -6998474114188967936, -110894925369151848448
Offset: 0

Views

Author

Seiichi Manyama, Aug 29 2022

Keywords

Links

Crossrefs

Cf. A356816, A356818.
Cf. A240165, A351736, A356812.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x*(exp(2*x)+1))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (-x)^k/(1-(2*k-1)*x)^(k+1)))
  • PARI
    a(n) = sum(k=0, n, (-1)^k*(2*k-1)^(n-k)*binomial(n, k));

Formula

G.f.: Sum_{k>=0} (-x)^k / (1 - (2*k-1)*x)^(k+1).
a(n) = Sum_{k=0..n} (-1)^k * (2*k-1)^(n-k) * binomial(n,k).

A351733 Expansion of e.g.f. exp( 2 * x * (exp(x) - 1) ).

Original entry on oeis.org

1, 0, 4, 6, 56, 250, 1812, 12614, 101040, 864882, 7988780, 78726142, 823897032, 9111774698, 106068603396, 1295153135670, 16538681229152, 220281968528098, 3053087839536732, 43941561067048430, 655501502129291640, 10118103843683127642
Offset: 0

Views

Author

Seiichi Manyama, May 20 2022

Keywords

Crossrefs

Cf. A052506, A351734.
Cf. A053489, A351736.

Programs

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

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} 2^k * Stirling2(n-k,k)/(n-k)!.

A354311 Expansion of e.g.f. exp( x/2 * (exp(2 * x) - 1) ).

Original entry on oeis.org

1, 0, 2, 6, 28, 160, 1056, 7784, 63568, 569664, 5542240, 58038112, 650045760, 7746901760, 97790608384, 1302349549440, 18235836899584, 267663541270528, 4107395264113152, 65739857693144576, 1095095457262013440, 18949711553467957248, 340036076121127395328
Offset: 0

Views

Author

Seiichi Manyama, May 23 2022

Keywords

Crossrefs

Cf. A052506, A354312.
Cf. A351733, A351736, A354309, A354313.

Programs

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

Formula

a(0) = 1; a(n) = Sum_{k=2..n} k * 2^(k-2) * binomial(n-1,k-1) * a(n-k).
a(n) = n! * Sum_{k=0..floor(n/2)} 2^(n-2*k) * Stirling2(n-k,k)/(n-k)!.

A367885 Expansion of e.g.f. 1/(1 - x * (exp(2*x) - 1)).

Original entry on oeis.org

1, 0, 4, 12, 128, 1040, 12672, 161728, 2481152, 41806080, 791613440, 16399944704, 371591995392, 9110211874816, 240670782291968, 6810264853463040, 205583847590985728, 6593508525460226048, 223913466256013918208, 8026367531323488993280
Offset: 0

Views

Author

Seiichi Manyama, Dec 04 2023

Keywords

Crossrefs

Cf. A052848, A367886.
Cf. A337038, A351736.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\2, 2^(n-k)*k!*stirling(n-k, k, 2)/(n-k)!);

Formula

a(0) = 1; a(n) = n * Sum_{k=2..n} 2^(k-1) * binomial(n-1,k-1) * a(n-k).
a(n) = n! * Sum_{k=0..floor(n/2)} 2^(n-k) * k! * Stirling2(n-k,k)/(n-k)!.

A362839 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} k^(n-j) * Stirling2(n-j,j)/(n-j)!.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 4, 3, 0, 1, 0, 6, 12, 16, 0, 1, 0, 8, 27, 80, 65, 0, 1, 0, 10, 48, 216, 560, 336, 0, 1, 0, 12, 75, 448, 2025, 4512, 1897, 0, 1, 0, 14, 108, 800, 5120, 21708, 40768, 11824, 0, 1, 0, 16, 147, 1296, 10625, 67584, 260253, 407808, 80145, 0
Offset: 0

Views

Author

Seiichi Manyama, May 05 2023

Keywords

Examples

			Square array begins:
  1,  1,   1,    1,    1,     1, ...
  0,  0,   0,    0,    0,     0, ...
  0,  2,   4,    6,    8,    10, ...
  0,  3,  12,   27,   48,    75, ...
  0, 16,  80,  216,  448,   800, ...
  0, 65, 560, 2025, 5120, 10625, ...

Crossrefs

Columns k=0..3 give: A000007, A052506, A351736, A351737.
Main diagonal gives A356806.
Cf. A362652.

Programs

  • PARI
    T(n, k) = n!*sum(j=0, n\2, k^(n-j)*stirling(n-j, j, 2)/(n-j)!);

Formula

E.g.f. of column k: exp(x * (exp(k * x) - 1)).
G.f. of column k: Sum_{j>=0} x^j / (1 - (k*j-1)*x)^(j+1).
T(n,k) = Sum_{j=0..n} (k*j-1)^(n-j) * binomial(n,j).
Showing 1-9 of 9 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.