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 1-4 of 4 results.

A075834 Coefficients of power series A(x) such that n-th term of A(x)^n = n! x^(n-1) for n > 0.

Original entry on oeis.org

1, 1, 1, 2, 7, 34, 206, 1476, 12123, 111866, 1143554, 12816572, 156217782, 2057246164, 29111150620, 440565923336, 7101696260883, 121489909224618, 2198572792193786, 41966290373704332, 842706170872913634, 17759399688526009020, 391929722837419044420
Offset: 0

Views

Author

Paul D. Hanna, Oct 14 2002, Jul 30 2008

Keywords

Comments

Also, number of stablized-interval-free permutations on [n] (see Callan link).
Coefficients in the series reversal of the asymptotic expansion of exp(-x)*Ei(x) for x -> inf, where Ei(x) is the exponential integral. - Vladimir Reshetnikov, Apr 24 2016

Examples

			At n=7, the 7th term of A(x)^7 is 7! x^6, as demonstrated by A(x)^7 = 1 + 7 x + 28 x^2 + 91 x^3 + 294 x^4 + 1092 x^5 + 5040 x^6 + 29093 x^7 + 203651 x^8 + ... .
A(x) = 1 + x + x^2 + 2*x^3 + 7*x^4 + 34*x^5 + 206*x^6 + ... = x/series_reversion(x + x^2 + 2*x^3 + 6*x^4 + 24*x^5 + 120*x^6 + ...).
Related expansions:
log(A(x)) = x + x^2/2 + 4*x^3/3 + 21*x^4/4 + 136*x^5/5 + 1030*x^6/6 + ...;
1 - x/(A(x) - 1) = x + x^2 + 4*x^3 + 21*x^4 + 136*x^5 + 1030*x^6 +...;
(d/dx)((A(x) - 1)/x) = 1 + 4*x + 21*x^2 + 136*x^3 + 1030*x^4 + ... .

Links

Crossrefs

Cf. A209881, A091063, A084938.
Cf. A003319.

Programs

  • Mathematica
    a = ConstantArray[0,20]; a[[1]]=1; a[[2]]=1; a[[3]]=2; Do[a[[n]] = (n-1)*a[[n-1]] + Sum[(j-1)*a[[j]]*a[[n-j]],{j,2,n-2}],{n,4,20}]; Flatten[{1,a}] (* Vaclav Kotesovec after David Callan, Feb 22 2014 *)
InverseSeries[Series[Exp[-x] ExpIntegralEi[x], {x, Infinity, 20}]][[3]] (* Vladimir Reshetnikov, Apr 24 2016 *)
  • PARI
    a(n)=if(n<0,0,if(n<=1,1,(n-1)*a(n-1)+sum(j=2,n-2,(j-1)*a(j)*a(n-j));))
  • PARI
    a(n)=Vec(x/serreverse(x*Ser(vector(n+1,k,(k-1)!))))[n+1] \\ Paul D. Hanna, Jul 09 2006
  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x/(1-x*deriv(A)/A));polcoeff(A,n)}
  • PARI
    {a(n)=local(F=1+x*O(x^n)); for(i=0,n,F=1+x*F+x^2*F*deriv(F)+x*O(x^n));polcoeff(1+x*F,n)} \\ Paul D. Hanna, Sep 02 2008

Formula

a(0)=a(1)=1, a(n) = (n-1)*a(n-1) + Sum_{j=2..n-2}(j-1)*a(j)*a(n-j), n >= 2. - David Callan
G.f.: A(x) = x/series_reversion(x*G(x)); G(x) = A(x*G(x)); A(x) = G(x/A(x)); where G(x) is the g.f. of the factorials (A000142). - Paul D. Hanna, Jul 09 2006
G.f.: A(x) = 1 + x/(1 - x*A'(x)/A(x)) = 1 + x/(1-x - x^2*d/dx((A(x) - 1)/x)).
G.f.: A(x) = 1 + x*F(x) where F(x) satisfies F(x) = 1 + x*F(x) + x^2*F(x)*F'(x) and F'(x) = d/dx F(x). - Paul D. Hanna, Sep 02 2008
a(n) ~ exp(-1) * n! * (1 - 1/n - 5/(2*n^2) - 32/(3*n^3) - 1643/(24*n^4) - 23017/(40*n^5) - 4215719/(720*n^6)). - Vaclav Kotesovec, Feb 22 2014
A003319(n+1) = coefficient of x^n in A(x)^n. - Michael Somos, Feb 23 2014

Extensions

More terms from David Wasserman, Jan 26 2005
Minor edits by Vaclav Kotesovec, Aug 01 2015

A305602 G.f. A(x) satisfies: A(x) = 1 + x*[d/dx 1/(1 - x*A(x)^2)].

Original entry on oeis.org

1, 1, 6, 54, 628, 8760, 140904, 2552151, 51243864, 1127982321, 26993774100, 697703846499, 19372450060296, 575205186725962, 18191422973198622, 610655961723782310, 21689599103526363600, 812832263931582168447, 32057155649057309677062, 1327393477257351399000744, 57581802198755959140129600
Offset: 0

Views

Author

Paul D. Hanna, Jun 05 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 54*x^3 + 628*x^4 + 8760*x^5 + 140904*x^6 + 2552151*x^7 + 51243864*x^8 + 1127982321*x^9 + 26993774100*x^10 + ...
such that A(x) = 1 + x*[d/dx 1/(1 - x*A(x)^2)].
RELATED SERIES.
A(x)^2 = 1 + 2*x + 13*x^2 + 120*x^3 + 1400*x^4 + 19424*x^5 + 309780*x^6 + 5559054*x^7 + 110623342*x^8 + 2415298374*x^9 + 57387784542*x^10 + ...
1/(1 - x*A(x)^2) = 1 + x + 3*x^2 + 18*x^3 + 157*x^4 + 1752*x^5 + 23484*x^6 + 364593*x^7 + 6405483*x^8 + 125331369*x^9 + 2699377410*x^10 + ...
exp( Integral A(x)^2 dx ) = 1 + x + 3*x^2/2! + 33*x^3/3! + 849*x^4/4! + 38061*x^5/5! + 2575611*x^6/6! + 242377533*x^7/7! + 30085188993*x^8/8! + ...
A'(x)/A(x) = 1 + 11*x + 145*x^2 + 2247*x^3 + 39461*x^4 + 768983*x^5 + 16409646*x^6 + 380013063*x^7 + 9487631035*x^8 + 254076973011*x^9 + ...

Links

Crossrefs

Cf. A305603, A305604, A209881, A305110.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=1+x*deriv(1/(1-x*A^2+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    {a(n) = my(A=[1], m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp( (m-1)*intformal(Ser(A)^2) ) * ((m-1) + 1 - Ser(A)) )[m] ); A[n+1]}
for(n=0, 25, print1(a(n), ", "))

Formula

O.g.f. A(x) satisfies:
(1) [x^n] exp( n * Integral A(x)^2 dx ) * (n + 1 - A(x)) = 0 for n > 0.
(2) A(x) = 1 + x*A(x)*(A(x) + 2*x*A'(x))/(1 - x*A(x)^2)^2.
a(n) ~ c * 2^n * n^(3/2) * n!, where c = 0.26934871195193907483980578... - Vaclav Kotesovec, Oct 06 2020

A305603 G.f. A(x) satisfies: A(x) = 1 + x*[d/dx 1/(1 - x*A(x)^3)].

Original entry on oeis.org

1, 1, 8, 102, 1712, 34785, 819384, 21810124, 645122272, 20957720148, 741260263600, 28350052179438, 1165931175542064, 51320048879474206, 2407857124657086480, 119990501174741855400, 6330579163195128292800, 352584892981590315935084, 20675941712941698695206368, 1273517057922072215818491064, 82210136955409063394289646720
Offset: 0

Views

Author

Paul D. Hanna, Jun 05 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 8*x^2 + 102*x^3 + 1712*x^4 + 34785*x^5 + 819384*x^6 + 21810124*x^7 + 645122272*x^8 + 20957720148*x^9 + 741260263600*x^10 + ...
such that A(x) = 1 + x*[d/dx 1/(1 - x*A(x)^3)].
RELATED SERIES.
A(x)^3 = 1 + 3*x + 27*x^2 + 355*x^3 + 5964*x^4 + 120021*x^5 + 2790794*x^6 + 73301427*x^7 + 2141393220*x^8 + 68800518492*x^9 + ...
1/(1 - x*A(x)^3) = 1 + x + 4*x^2 + 34*x^3 + 428*x^4 + 6957*x^5 + 136564*x^6 + 3115732*x^7 + 80640284*x^8 + 2328635572*x^9 + ...
A'(x)/A(x) = 1 + 15*x + 283*x^2 + 6343*x^3 + 162076*x^4 + 4614153*x^5 + 144287466*x^6 + 4908441479*x^7 + 180383821348*x^8 + 7122692545660*x^9 + ...

Links

Crossrefs

Cf. A305602, A305604, A209881, A305110.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=1+x*deriv(1/(1-x*A^3+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    {a(n) = my(A=[1], m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp( (m-1)*intformal(Ser(A)^3) ) * ((m-1) + 1 - Ser(A)) )[m] ); A[n+1]}
for(n=0, 25, print1(a(n), ", "))

Formula

O.g.f. A(x) satisfies:
(1) [x^n] exp( n * Integral A(x)^3 dx ) * (n + 1 - A(x)) = 0 for n > 0.
(2) A(x) = 1 + x*A(x)^2*(A(x) + 3*x*A'(x))/(1 - x*A(x)^3)^2.
a(n) ~ c * 3^n * n^(4/3) * n!, where c = 0.1925904251831569484470022... - Vaclav Kotesovec, Oct 06 2020

A305604 G.f. A(x) satisfies: A(x) = 1 + x*[d/dx 1/(1 - x*A(x)^4)].

Original entry on oeis.org

1, 1, 10, 165, 3620, 96600, 2996586, 105222740, 4110953640, 176563668420, 8262487393200, 418357209052255, 22790291724038580, 1329373540730613885, 82689132241752729840, 5464831115387029530825, 382492359049214499350160, 28269306654212844076331775, 2200386909130514596330158960, 179936864840741390285785502825, 15424543660650808111807757322400
Offset: 0

Views

Author

Paul D. Hanna, Jun 05 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 10*x^2 + 165*x^3 + 3620*x^4 + 96600*x^5 + 2996586*x^6 + 105222740*x^7 + 4110953640*x^8 + 176563668420*x^9 + ...
such that A(x) = 1 + x*[d/dx 1/(1 - x*A(x)^4)].
RELATED SERIES.
A(x)^4 = 1 + 4*x + 46*x^2 + 784*x^3 + 17181*x^4 + 452860*x^5 + 13831594*x^6 + 478200572*x^7 + 18418253542*x^8 + 781180290204*x^9 + ...
1/(1 - x*A(x)^4) = 1 + x + 5*x^2 + 55*x^3 + 905*x^4 + 19320*x^5 + 499431*x^6 + 15031820*x^7 + 513869205*x^8 + 19618185380*x^9 + ...
A'(x)/A(x) = 1 + 19*x + 466*x^2 + 13659*x^3 + 457926*x^4 + 17142730*x^5 + 706064549*x^6 + 31677960427*x^7 + 1537022113117*x^8 + ...

Links

Crossrefs

Cf. A305602, A305603, A209881, A305110.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=1+x*deriv(1/(1 - x*A^4+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    {a(n) = my(A=[1], m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp( (m-1)*intformal(Ser(A)^4) ) * ((m-1) + 1 - Ser(A)) )[m] ); A[n+1]}
for(n=0, 25, print1(a(n), ", "))

Formula

O.g.f. A(x) satisfies:
(1) [x^n] exp( n * Integral A(x)^4 dx ) * (n + 1 - A(x)) = 0 for n > 0.
(2) A(x) = 1 + x*A(x)^3*(A(x) + 4*x*A'(x))/(1 - x*A(x)^4)^2.
a(n) ~ c * 4^n * n^(5/4) * n!, where c = 0.147639333661398142298711... - Vaclav Kotesovec, Oct 06 2020
Showing 1-4 of 4 results.

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