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 21 results. Next

A365438 E.g.f. satisfies A(x) = 1 - log(1 - x*A(x)^2)/A(x).

Original entry on oeis.org

1, 1, 3, 20, 216, 3214, 60940, 1405088, 38165904, 1193631360, 42244603368, 1669171435392, 72834612333120, 3478615044283872, 180496518526631424, 10110668949900238848, 608110470593816945664, 39086875354688578492416, 2673826803093451383429120
Offset: 0

Views

Author

Seiichi Manyama, Nov 08 2023

Keywords

Crossrefs

Cf. A138013, A365546.

Programs

  • PARI
    a(n) = sum(k=0, n, (2*n-k)!/(2*n-2*k+1)!*abs(stirling(n, k, 1)));

Formula

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

A371117 E.g.f. satisfies A(x) = 1 - x*log(1 - x*A(x)).

Original entry on oeis.org

1, 0, 2, 3, 32, 210, 2184, 26460, 373344, 6150816, 113958720, 2362345920, 54094694400, 1355708296800, 36926213869440, 1085886303989760, 34291129916574720, 1157362522046277120, 41576054625791078400, 1583864892141097098240, 63779322541075124428800
Offset: 0

Views

Author

Seiichi Manyama, Mar 11 2024

Keywords

Crossrefs

Cf. A138013, A371118.
Cf. A371115.

Programs

  • Mathematica
    nmax = 20; A[] = 0; Do[A[x] = 1 - x*Log[1 - x*A[x]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] * Range[0,nmax]! (* Vaclav Kotesovec, Mar 11 2024 *)
  • PARI
    a(n) = n!*sum(k=0, n\2, abs(stirling(n-k, k, 1))/(n-2*k+1)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} |Stirling1(n-k,k)|/(n-2*k+1)!.
a(n) ~ sqrt(2 - r*(2*r+1)) * n^(n-1) / (exp(n) * r^n), where r = 0.4599065470184992266076522060382204730855199647380... is the root of the equation 1/r + 2*r*log(r) = 1+r. - Vaclav Kotesovec, Mar 11 2024

A177380 E.g.f. satisfies: A(x) = 1+x + x*log(A(x)).

Original entry on oeis.org

1, 1, 2, 3, -4, -50, -36, 2058, 10800, -131616, -1975680, 7741800, 417480480, 1307617584, -101626746144, -1284067345680, 25419094122240, 791333924647680, -3900043588999680, -472446912421801728, -3183064994777932800
Offset: 0

Views

Author

Paul D. Hanna, May 14 2010

Keywords

Comments

The signs have a complex structure; are they periodic after some point?

Examples

			E.g.f: A(x) = 1 + x + 2*x^2/2! + 3*x^3/3! - 4*x^4/4! - 50*x^5/5! +...
log(A(x)) = 2*x/2! + 3*x^2/3! - 4*x^3/4! - 50*x^4/5! - 36*x^5/5! +...
...
Coefficients in the initial powers of A(x) begin:
[1,(1),(1), 1/2, -1/6, -5/12, -1/20, 49/120, 15/56, -457/1260,...];
[1, 2,(3),(3), 5/3, -1/6, -61/60, -17/60, 272/315, 451/630,...];
[1, 3, 6,(17/2),(17/2), 21/4, 3/5, -83/40, -187/168, 115/84,...];
[1, 4, 10, 18,(73/3),(73/3), 163/10, 131/30, -261/70, -1093/315,...];
[1, 5, 15, 65/2, 325/6,(847/12),(847/12), 1205/24, 9551/504,...];
[1, 6, 21, 53, 104, 327/2,(4139/20),(4139/20), 6469/42, 7414/105,...];
[1, 7, 28, 161/2, 1085/6, 3955/12, 4949/10,(24477/40),(24477/40),...];
[1, 8, 36, 116, 878/3, 1810/3, 15569/15, 7509/5,(114760/63),(114760/63), ...]; ...
where the coefficients in parenthesis illustrate the property
that the coefficients of x^n and x^(n+1) in A(x)^n are equal:
[x^n] A(x)^n = [x^(n+1)] A(x)^n = A138013(n)/(n-1)!,
where G(x) = e.g.f. of A138013 begins:
G(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 146*x^4/4! + 1694*x^5/5! + ...
and satisfies: exp(1 - G(x)) = 1 - x*G(x).

Links

Crossrefs

Cf. A177379, A138013, A038037, A238274.

Programs

  • Mathematica
    CoefficientList[1+InverseSeries[Series[x/(1 + Log[1+x]), {x, 0, 20}], x],x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 11 2014 *)
  • PARI
    {a(n)=n!*polcoeff(1+serreverse(x/(1+log(1+x+x*O(x^n)))),n)}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+x+x*log(A+O(x^n)));n!*polcoeff(A,n)}

Formula

E.g.f.: A(x) = 1 + Series_Reversion( x/(1 + log(1+x)) ).
...
Let G(x) = e.g.f. of A138013, then G(x) and A(x) satisfy:
(1) [x^n] A(x)^n = [x^(n+1)] A(x)^n = A138013(n)/(n-1)! for n>=1;
(2) A(x/(1 - x*G(x))) = 1/(1 - x*G(x));
(3) G(x) = 1 - log(1 - x*G(x)) = Series_Reversion(x/(1-log(1-x)))/x.
...
Let F(x) = e.g.f. of A177379, then F(x) and A(x) satisfy:
(4) [x^n] A(x)^(n+1)/(n+1) = A177379(n)/n! for n>=0;
(5) A(x*F(x)) = F(x) and F(x/A(x)) = A(x);
(6) F(x) = 1/(1 - x*G(x)) = 1/(1 - Series_Reversion(x/(1-log(1-x)))).
Lim sup n->infinity (|a(n)|/n!)^(1/n) = abs(LambertW(-1)) = 1.3745570107437... (see A238274). - Vaclav Kotesovec, Jan 11 2014

A185221 E.g.f. is solution to y = 1 + log(1 + x*y) in powers of x.

Original entry on oeis.org

1, 1, 1, -1, -10, -6, 294, 1350, -14624, -197568, 703800, 34790040, 100585968, -7259053296, -85604489712, 1588693382640, 46549054391040, -216669088277760, -24865626969568512, -159153249738896640, 13379663931502199040
Offset: 0

Views

Author

Michael Somos, Jan 24 2012

Keywords

Examples

			y = 1 + x + 1/2*x^2 - 1/6*x^3 - 5/12*x^4 - 1/20*x^5 + 49/120*x^6 + 15/56*x^7 + ...

Links

Crossrefs

Cf. A177380, A238274.
Cf. A097718, A138013.

Programs

  • Maxima
    a(n):=if n=0 then 1 else sum(binomial(n+k+1,n) * sum((-1)^(j) * binomial(k+1,j) * sum((-1)^i * i! * binomial(j+i-1,j-1) * stirling1(n,i), i,1,n), j,1,k+1), k,0,n) / (n+1); /* Vladimir Kruchinin, Mar 29 2013 */
  • PARI
    {a(n) = local(A); if( n<0, 0, A = 1 + O(x); for( k=1, n, A = 1 + log(1 + x * A)); n! * polcoeff( A, n))}

Formula

E.g.f. is solution to y = y' * (1 - x + x*y).
a(n) = sum(k=0..n, binomial(n+k+1,n) * sum(j=1..k+1, (-1)^(j) * binomial(k+1,j) * sum(i=1..n, (-1)^i * i! * binomial(j+i-1,j-1) * stirling1(n,i)))) / (n+1), n>0, a(0)=1. [Vladimir Kruchinin, Mar 29 2013]
Lim sup n->infinity (|a(n)|/n!)^(1/n) = abs(LambertW(-1)) = 1.37455701074370748653... (see A238274). - Vaclav Kotesovec, Feb 24 2014
a(n) = n! * Sum_{k=0..n} Stirling1(n,k)/(n-k+1)!. - Seiichi Manyama, Nov 07 2023

A367080 E.g.f. satisfies A(x) = 1 - log(1 - x*A(x)^2).

Original entry on oeis.org

1, 1, 5, 50, 766, 15914, 418548, 13337624, 499600848, 21516318360, 1047593782440, 56903921842272, 3411723783002016, 223803339516120480, 15944855840879771232, 1226078375934824887680, 101209861891840507123200, 8926972851724904613537792
Offset: 0

Views

Author

Seiichi Manyama, Nov 07 2023

Keywords

Crossrefs

Cf. A138013, A367152.
Cf. A367078, A367153.

Programs

  • Mathematica
    Table[(2*n)! * Sum[Abs[StirlingS1[n,k]]/(2*n-k+1)!, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Nov 07 2023 *)
  • PARI
    a(n) = (2*n)!*sum(k=0, n, abs(stirling(n, k, 1))/(2*n-k+1)!);

Formula

a(n) = (2*n)! * Sum_{k=0..n} |Stirling1(n,k)|/(2*n-k+1)!.
a(n) ~ (-2 - LambertW(-1, -2*exp(-3)))^(n+1) * (-LambertW(-1, -2*exp(-3)))^n * n^(n-1) / (sqrt(-2 - 2*LambertW(-1, -2*exp(-3))) * exp(n)). - Vaclav Kotesovec, Nov 07 2023

A367152 E.g.f. satisfies A(x) = 1 - log(1 - x*A(x)^3).

Original entry on oeis.org

1, 1, 7, 101, 2250, 68184, 2619822, 122071704, 6689791392, 421670267136, 30055781201520, 2390512621714656, 209893714832795760, 20165895195283566000, 2104433775967024226592, 237043144515185017456320, 28664975599576485530851584, 3704019298858867019823244800
Offset: 0

Views

Author

Seiichi Manyama, Nov 07 2023

Keywords

Crossrefs

Cf. A138013, A367080.

Programs

  • Mathematica
    Table[(3*n)! * Sum[Abs[StirlingS1[n,k]]/(3*n-k+1)!, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Nov 07 2023 *)
  • PARI
    a(n) = (3*n)!*sum(k=0, n, abs(stirling(n, k, 1))/(3*n-k+1)!);

Formula

a(n) = (3*n)! * Sum_{k=0..n} |Stirling1(n,k)|/(3*n-k+1)!.
a(n) ~ (-3 - LambertW(-1, -3*exp(-4)))^(2*n+1) * (-LambertW(-1, -3*exp(-4)))^n * n^(n-1) / (sqrt(-3 - 3*LambertW(-1, -3*exp(-4))) * exp(n)). - Vaclav Kotesovec, Nov 07 2023

A367159 E.g.f. satisfies A(x) = 1 - A(x)^2 * log(1 - x*A(x)).

Original entry on oeis.org

1, 1, 7, 95, 1954, 54244, 1901560, 80648658, 4016874920, 229881369768, 14866341101064, 1072223706468672, 85337672738960736, 7429736462231570304, 702426961910810154624, 71667022709644235679120, 7848761844632669045606016, 918383128248130459272478080
Offset: 0

Views

Author

Seiichi Manyama, Nov 07 2023

Keywords

Crossrefs

Cf. A052802, A138013, A367160.

Programs

  • PARI
    a(n) = sum(k=0, n, (n+2*k)!/(n+k+1)!*abs(stirling(n, k, 1)));

Formula

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

A365546 E.g.f. satisfies A(x) = 1 - log(1 - x*A(x)^3)/A(x)^2.

Original entry on oeis.org

1, 1, 3, 23, 298, 5314, 120776, 3341568, 108992472, 4095073848, 174169888536, 8272115427432, 433956083676336, 24921123498835056, 1555004372522100384, 104757005524567577088, 7578056156152486855680, 585874671534300791384064
Offset: 0

Views

Author

Seiichi Manyama, Nov 08 2023

Keywords

Crossrefs

Cf. A138013, A365438.

Programs

  • PARI
    a(n) = sum(k=0, n, (3*n-2*k)!/(3*n-3*k+1)!*abs(stirling(n, k, 1)));

Formula

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

A371118 E.g.f. satisfies A(x) = 1 - x^2*log(1 - x*A(x)).

Original entry on oeis.org

1, 0, 0, 6, 12, 40, 900, 8568, 80640, 1170720, 19625760, 335079360, 6259524480, 132897697920, 3078950434560, 75934287475200, 2010354982272000, 57241482249369600, 1735591000023336960, 55694476130213652480, 1889613850762113638400
Offset: 0

Views

Author

Seiichi Manyama, Mar 11 2024

Keywords

Crossrefs

Cf. A138013, A371117.
Cf. A371116.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\3, abs(stirling(n-2*k, k, 1))/(n-3*k+1)!);

Formula

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

A377325 E.g.f. satisfies A(x) = 1 - log(1 - x*A(x))/A(x).

Original entry on oeis.org

1, 1, 1, 5, 28, 244, 2566, 33438, 508544, 8926944, 176989488, 3917823216, 95719041408, 2559130965312, 74312569125744, 2329169772108528, 78371469374088960, 2817744760964392704, 107807187260426164992, 4373419962377871956736, 187507942522161269068800
Offset: 0

Views

Author

Seiichi Manyama, Oct 24 2024

Keywords

Crossrefs

Cf. A052802, A138013, A367159.
Cf. A365438, A377323.

Programs

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

Formula

a(n) = Sum_{k=0..floor((n+1)/2)} (n-k)!/(n-2*k+1)! * |Stirling1(n,k)|.
Showing 1-10 of 21 results. Next

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