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.

A161630 E.g.f. satisfies: A(x) = exp( x/(1 - x*A(x)) ).

Original entry on oeis.org

1, 1, 3, 19, 181, 2321, 37501, 731935, 16758393, 440525377, 13077834841, 432796650551, 15799794395749, 630773263606513, 27339525297079269, 1278550150117141231, 64171287394646697841, 3440711053857464325377
Offset: 0

Views

Author

Paul D. Hanna, Jun 17 2009

Keywords

Examples

			E.g.f: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 181*x^4/4! + 2321*x^5/5! +...
log(A(x))/x = 1 + x*A(x) + x^2*A(x)^2 + x^3*A(x)^3 + x^4*A(x)^4 +...

Links

Crossrefs

Cf. A161633 (e.g.f. = log(A(x))/x).
Cf. A212722, A212917, A245265, A125500.

Programs

  • Mathematica
    Table[Sum[n! * (n-k+1)^(k-1)/k! * Binomial[n-1,n-k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jan 10 2014 *)
  • PARI
    {a(n,m=1)=if(n==0,1,sum(k=0,n,n!/k!*m*(n-k+m)^(k-1)*binomial(n-1,n-k)))}
  • PARI
    {a(n,m=1)=local(A=1+x+x*O(x^n));for(i=1,n,A=exp(x/(1-x*A)));n!*polcoeff(A^m,n)}

Formula

a(n) = Sum_{k=0..n} n! * (n-k+1)^(k-1)/k! * C(n-1,n-k).
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
a(n,m) = Sum_{k=0..n} n! * m*(n-k+m)^(k-1)/k! * C(n-1,n-k).
E.g.f. satisfies: A(x) = exp(x) * A(x)^(x*A(x)). - Paul D. Hanna, Aug 02 2013
a(n) ~ n^(n-1) * (1+2*c)^(n+1/2) / (sqrt(1+c) * 2^(2*n+2) * exp(n) * c^(2*n+3/2)), where c = LambertW(1/2) = 0.351733711249195826... (see A202356). - Vaclav Kotesovec, Jan 10 2014

A212722 E.g.f. satisfies: A(x) = exp( x/(1 - x*A(x)^2) ).

Original entry on oeis.org

1, 1, 3, 25, 313, 5341, 115651, 3036517, 93767185, 3330162073, 133737097411, 5992748728561, 296433923379529, 16044427276953973, 943207466055927619, 59848531677741706621, 4076826825898115406241, 296742863575079244130225
Offset: 0

Views

Author

Paul D. Hanna, May 25 2012

Keywords

Examples

			E.g.f: A(x) = 1 + x + 3*x^2/2! + 25*x^3/3! + 313*x^4/4! + 5341*x^5/5! + ...
such that, by definition:
log(A(x))/x = 1 + x*A(x)^2 + x^2*A(x)^4 + x^3*A(x)^6 + x^4*A(x)^8 + ...
Related expansions:
log(A(x)) = x/(1-x*A(x)^2) = x + 2*x^2/2! + 18*x^3/3! + 216*x^4/4! + 3640*x^5/5! + 78000*x^6/6! + 2032464*x^7/7! + 62400128*x^8/8! + ... + n*A366232(n-1)*x^n/n! + ...
A(x)^2 = 1 + 2*x + 8*x^2/2! + 68*x^3/3! + 880*x^4/4! + 15312*x^5/5! + 336064*x^6/6! +...
A(x)^4 = 1 + 4*x + 24*x^2/2! + 232*x^3/3! + 3232*x^4/4! + 59104*x^5/5! + 1343296*x^6/6! +...

Links

Crossrefs

Cf. A161630, A212917, A245265.
Cf. A366232 (log).

Programs

  • Mathematica
    Table[Sum[n! * (1 + 2*(n-k))^(k-1)/k! * Binomial[n-1,n-k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 15 2014 *)
  • PARI
    {a(n, m=1)=if(n==0, 1, sum(k=0, n, n!/k!*m*(m+2*(n-k))^(k-1)*binomial(n-1, n-k)))}
  • PARI
    {a(n, m=1)=local(A=1+x); for(i=1, n, A=exp(x/(1-x*A^2+x*O(x^n)))); n!*polcoeff(A^m, n)}
for(n=0,21,print1(a(n),", "))

Formula

a(n) = Sum_{k=0..n} n! * (1 + 2*(n-k))^(k-1)/k! * C(n-1,n-k).
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
a(n,m) = Sum_{k=0..n} n! * m*(m + 2*(n-k))^(k-1)/k! * C(n-1,n-k).
a(n) ~ n^(n-1) * (1+1/(2*c))^(n+1/2) / (2*sqrt(1+c) * exp(n) * c^n), where c = LambertW(1/sqrt(2)) = 0.450600515864833072257... . - Vaclav Kotesovec, Jul 15 2014

A212917 E.g.f. satisfies: A(x) = exp( x/(1 - x*A(x)^3) ).

Original entry on oeis.org

1, 1, 3, 31, 469, 9681, 254701, 8131999, 305626329, 13218345793, 646712664121, 35315446759671, 2129341219106773, 140506900034640049, 10071589943109973461, 779311468200041101711, 64742128053980794659121, 5747587082198264156035329, 543023929087191507383612785
Offset: 0

Views

Author

Paul D. Hanna, May 30 2012

Keywords

Comments

From Vaclav Kotesovec, Jul 15 2014: (Start)
Generally, if e.g.f. satisfies: A(x) = exp(x/(1-x*A(x)^p)), p>=1, then
r = 4*LambertW(sqrt(p)/2)^2 / (p*(1+2*LambertW(sqrt(p)/2))),
A(r) = (sqrt(p)/(2*LambertW(sqrt(p)/2)))^(2/p),
a(n) ~ p^(n-1+1/p) * (1+2*LambertW(sqrt(p)/2))^(n+1/2) * n^(n-1) / (sqrt(1+LambertW(sqrt(p)/2)) * exp(n) * 2^(2*n+2/p) * LambertW(sqrt(p)/2)^(2*n+2/p-1/2)).
(End)

Examples

			E.g.f: A(x) = 1 + x + 3*x^2/2! + 31*x^3/3! + 469*x^4/4! + 9681*x^5/5! + ...
such that, by definition:
log(A(x))/x = 1 + x*A(x)^3 + x^2*A(x)^6 + x^3*A(x)^9 + x^4*A(x)^12 + ...
Related expansions:
log(A(x)) = x/(1-x*A(x)^3) = x + 2*x^2/2! + 24*x^3/3! + 348*x^4/4! + 7140*x^5/5! + 186750*x^6/6! + ... + n*A366233(n-1)*x^n/n! + ...
A(x)^3 = 1 + 3*x + 15*x^2/2! + 153*x^3/3! + 2421*x^4/4! + 51363*x^5/5! + 1375029*x^6/6! + ...
A(x)^6 = 1 + 6*x + 48*x^2/2! + 576*x^3/3! + 9864*x^4/4! + 221256*x^5/5! + 6156756*x^6/6! + ...

Links

Crossrefs

Cf. A161630, A212722, A245265.
Cf. A366233 (log).

Programs

  • Mathematica
    Table[Sum[n! * (1 + 3*(n-k))^(k-1)/k! * Binomial[n-1,n-k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 15 2014 *)
  • PARI
    {a(n, m=1)=if(n==0, 1, sum(k=0, n, n!/k!*m*(m+3*(n-k))^(k-1)*binomial(n-1, n-k)))}
  • PARI
    {a(n, m=1)=local(A=1+x); for(i=1, n, A=exp(x/(1-x*A^3+x*O(x^n)))); n!*polcoeff(A^m, n)}
for(n=0, 21, print1(a(n), ", "))

Formula

a(n) = Sum_{k=0..n} n! * (1 + 3*(n-k))^(k-1)/k! * C(n-1,n-k).
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
a(n,m) = Sum_{k=0..n} n! * m*(m + 3*(n-k))^(k-1)/k! * C(n-1,n-k).
a(n) ~ 3^(n-2/3) * n^(n-1) * (1+2*c)^(n+1/2) / (sqrt(1+c) * 2^(2*n+2/3) * exp(n) * c^(2*n+1/6)), where c = LambertW(sqrt(3)/2) = 0.5166154518588324282494... . - Vaclav Kotesovec, Jul 15 2014

A366234 Expansion of e.g.f. A(x) satisfying A(x) = 1 + x*A(x) * exp(4*x*A(x)).

Original entry on oeis.org

1, 1, 10, 126, 2392, 60600, 1916304, 72917488, 3246171520, 165609099648, 9529240349440, 610657739172096, 43136025287678976, 3330356645773880320, 279024535906794539008, 25214258236430338160640, 2444656672390982922502144, 253144081975231633923342336
Offset: 0

Views

Author

Paul D. Hanna, Oct 05 2023

Keywords

Comments

Related identities which hold formally for all Maclaurin series F(x):
(1) F(x) = (1/x) * Sum_{n>=1} n^(n-1) * x^n/n! * F(x)^n * exp(-n*x*F(x)),
(2) F(x) = (2/x) * Sum_{n>=1} n*(n+1)^(n-2) * x^n/n! * F(x)^n * exp(-(n+1)*x*F(x)),
(3) F(x) = (3/x) * Sum_{n>=1} n*(n+2)^(n-2) * x^n/n! * F(x)^n * exp(-(n+2)*x*F(x)),
(4) F(x) = (4/x) * Sum_{n>=1} n*(n+3)^(n-2) * x^n/n! * F(x)^n * exp(-(n+3)*x*F(x)),
(5) F(x) = (k/x) * Sum_{n>=1} n*(n+k-1)^(n-2) * x^n/n! * F(x)^n * exp(-(n+k-1)*x*F(x)) for all fixed nonzero k.

Examples

			E.g.f.: A(x) = 1 + x + 10*x^2/2! + 126*x^3/3! + 2392*x^4/4! + 60600*x^5/5! + 1916304*x^6/6! + 72917488*x^7/7! + 3246171520*x^8/8! + ...
where A(x) satisfies A(x) = 1 + x*A(x) * exp(4*x*A(x))
also
A(x) = 1 + 1^0*x*A(x)*exp(+3*x*A(x))/1! + 2^1*x^2*A(x)^2*exp(+2*x*A(x))/2! + 3^2*x^3*A(x)^3*exp(+1*x*A(x))/3! + 4^3*x^4*A(x)^4*exp(-0*x*A(x))/4! + 5^4*x^5*A(x)^5*exp(-1*x*A(x))/5! + 6^5*x^6*A(x)^6*exp(-2*x*A(x))/6! + ...
and
A(x) = 1 + 5*1*5^(-1)*x*A(x)*exp(-1*x*A(x))/1! + 5*2*6^0*x^2*A(x)^2*exp(-2*x*A(x))/2! + 5*3*7^1*x^3*A(x)^3*exp(-3*x*A(x))/3! + 5*4*8^2*x^4*A(x)^4*exp(-4*x*A(x))/4! + 5*5*9^3*x^5*A(x)^5*exp(-5*x*A(x))/5! + ...
RELATED SERIES.
exp(x*A(x)) = 1 + x + 3*x^2/2! + 37*x^3/3! + 649*x^4/4! + 15461*x^5/5! + 471571*x^6/6! + ... + A245265(n)*x^n/n! + ...

Crossrefs

Cf. A161633, A366232, A366233, A366235.
Cf. A365774 (dual), A245265 (exp(x*A(x))).

Programs

  • Mathematica
    nmax = 20; A[] = 0; Do[A[x] = 1 + x*A[x] * E^(4*x*A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] * Range[0,nmax]! (* Vaclav Kotesovec, Oct 06 2023 *)
  • PARI
    /* a(n,m) = coefficient of x^n/n! in A(x)^m, here at m = 1 */
{a(n, m=1) = n!*sum(k=0, n, binomial(n+m, n-k)*m/(n+m) * 4^k * (n-k)^k/k!)}
for(n=0,20,print1(a(n),", "))
  • PARI
    {a(n) = my(A = (1/x) * serreverse( x/(1 + x*exp(4*x +O(x^(n+2)))) )); n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

Formula

a(n) = n! * Sum_{k=0..n} binomial(n+1, n-k)/(n+1) * 4^k * (n-k)^k / k!.
Let A(x)^m = Sum_{n>=0} a(n,m) * x^n/n! then a(n,m) = n!*Sum_{k=0..n} binomial(n+m, n-k)*m/(n+m) * 4^k * (n-k)^k/k!.
E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = 1 + x*A(x) * exp(4*x*A(x)).
(2) A(x) = (1/x) * Series_Reversion( x/(1 + x*exp(4*x)) ).
(3) A( x/(1 + x*exp(4*x)) ) = 1 + x*exp(4*x).
(4) A(x) = 1 + (m+1) * Sum_{n>=1} n*(n+m)^(n-2) * x^n/n! * A(x)^n * exp(-(n+m-4)*x*A(x)) for all fixed nonnegative m.
(4.a) A(x) = 1 + Sum_{n>=1} n^(n-1) * x^n/n! * A(x)^n * exp(-(n-4)*x*A(x)).
(4.b) A(x) = 1 + 2 * Sum_{n>=1} n*(n+1)^(n-2) * x^n/n! * A(x)^n * exp(-(n-3)*x*A(x)).
(4.c) A(x) = 1 + 3 * Sum_{n>=1} n*(n+2)^(n-2) * x^n/n! * A(x)^n * exp(-(n-2)*x*A(x)).
(4.d) A(x) = 1 + 4 * Sum_{n>=1} n*(n+3)^(n-2) * x^n/n! * A(x)^n * exp(-(n-1)*x*A(x)).
(4.e) A(x) = 1 + 5 * Sum_{n>=1} n*(n+4)^(n-2) * x^n/n! * A(x)^n * exp(-n*x*A(x)).
(4.f) A(x) = 1 + 6 * Sum_{n>=1} n*(n+5)^(n-2) * x^n/n! * A(x)^n * exp(-(n+1)*x*A(x)).
a(n) ~ (1 + 2*LambertW(1))^(n + 3/2) * n^(n-1) / (4 * sqrt(1 + LambertW(1)) * exp(n) * LambertW(1)^(2*n + 3/2)). - Vaclav Kotesovec, Oct 06 2023
Showing 1-4 of 4 results.

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