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 1-6 of 6 results.

A207833 E.g.f.: T(T(x)), where T(x) is the e.g.f. for labeled rooted trees, A000169.

Original entry on oeis.org

1, 4, 30, 332, 4880, 89742, 1986124, 51471800, 1530489744, 51395228090, 1924687118684, 79553145323940, 3598161485778808, 176797212122233094, 9378715234039802340, 534259395682874552048, 32528761111972930621472, 2108146039402630977388530, 144899759883703796130871468, 10528261771566724089621962780
Offset: 1

Views

Author

N. J. A. Sloane, Feb 20 2012

Keywords

Comments

Exponential series reversal gives A185298 with alternating signs: 1, -4, 18, -92, 520, ... . - Vladimir Reshetnikov, Aug 04 2019

Examples

			E.g.f.: A(x) = x + 4*x^2/2! + 30*x^3/3! + 332*x^4/4! + 4880*x^5/5! +...
Euler's tree function T(x) satisfies: T(x/exp(x)) = x, and begins:
T(x) = x + 2*x^2/2! + 3^2*x^3/3! + 4^3*x^4/4! + 5^4*x^5/5! +...+ A000169(n)*x^n/n! +...
where e.g.f. A(x) = T(T(x)).

Links

Crossrefs

Cf. A000169, A185298, A227278, A227176.

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[t,{x,0,nn}]],x] (* Geoffrey Critzer, Sep 16 2012 *)
Rest[CoefficientList[Series[-LambertW[LambertW[-x]], {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Feb 24 2014 *)
  • PARI
    {a(n)=if(n==0||n==1, 1, n^(n-1)-sum(k=1, n-1, (-1)^(n-k)*binomial(n, k)*k^(n-k)*a(k)))} \\ Paul D. Hanna, Nov 21 2012

Formula

a(n) = 1/n * Sum_{k=1..n} C(n,k)*k^k*n^(n-k). [Vladimir Kruchinin, Sep 24 2012]
a(n) = n^(n-1) - Sum_{k=1..n-1} (-1)^(n-k) * C(n, k) * k^(n-k) * a(k) for n>1 with a(1)=1. - Paul D. Hanna, Nov 21 2012
E.g.f. A(x) satisfies: A(x) = Sum_{n>=1} n^(n-1)*T(x)^n/n!, by definition.
E.g.f. A(x) satisfies: A(x/exp(x)) = T(x) = Sum_{n>=1} n^(n-1)*x^n/n!. - Paul D. Hanna, Jul 04 2013
a(n) ~ n^(n-1) * exp(n*exp(-1)) / sqrt(1-exp(-1)). - Vaclav Kotesovec, Feb 24 2014

A274391 Table of coefficients in functions that satisfy W_n(x) = W_{n-1}(x)^W_n(x), with W_0(x) = exp(x), as read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 16, 1, 1, 1, 7, 43, 125, 1, 1, 1, 9, 82, 525, 1296, 1, 1, 1, 11, 133, 1345, 8321, 16807, 1, 1, 1, 13, 196, 2729, 28396, 162463, 262144, 1, 1, 1, 15, 271, 4821, 71721, 734149, 3774513, 4782969, 1, 1, 1, 17, 358, 7765, 151376, 2300485, 22485898, 101808185, 100000000, 1, 1, 1, 19, 457, 11705, 283321, 5787931, 87194689, 796769201, 3129525793, 2357947691, 1, 1, 1, 21, 568, 16785, 486396, 12567187, 261066156, 3815719969, 32084546824, 108063152091, 61917364224, 1, 1, 1, 23, 691, 23149, 782321, 24539593, 656778529, 13577077401, 189440927857, 1447917011461, 4143297446729, 1792160394037, 1, 1, 1, 25, 826, 30941, 1195696, 44223529, 1457297878, 39536713209, 800175234736, 10525328121221, 72411962077126, 174723134310277, 56693912375296, 1
Offset: 0

Views

Author

Paul D. Hanna, Jun 19 2016

Keywords

Comments

The e.g.f. of each row is an infinite exponential tetration of the e.g.f. of the prior row: W_{n+1}(x) = W_n(x)^W_n(x)^W_n(x)^..., starting with exp(x) as the e.g.f. of row zero. All of these row functions may be expressed in terms of the LambertW(x) function.

Examples

			This table begins:
1, 1,  1,   1,     1,       1,         1,          1,            1, ...;
1, 1,  3,  16,   125,    1296,     16807,     262144,      4782969, ...;
1, 1,  5,  43,   525,    8321,    162463,    3774513,    101808185, ...;
1, 1,  7,  82,  1345,   28396,    734149,   22485898,    796769201, ...;
1, 1,  9, 133,  2729,   71721,   2300485,   87194689,   3815719969, ...;
1, 1, 11, 196,  4821,  151376,   5787931,  261066156,  13577077401, ...;
1, 1, 13, 271,  7765,  283321,  12567187,  656778529,  39536713209, ...;
1, 1, 15, 358, 11705,  486396,  24539593, 1457297878,  99609347825, ...;
1, 1, 17, 457, 16785,  782321,  44223529, 2940281793, 224869459201, ...;
1, 1, 19, 568, 23149, 1195696,  74840815, 5506111864, 465734919289, ...;
1, 1, 21, 691, 30941, 1754001, 120403111, 9709554961, 899836571001, ...;
...
in which the e.g.f. of row n equals W_n(x) = exp( T^n(x) ), where T^n(x) is the n-th iteration of the Euler tree function T(x).
The row functions begin:
W_0(x) = 1 + x + x^2/2! + x^3/3! + x^4/4! + x^5/5! + x^6/6! +...;
W_1(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! + 16807*x^6/6! + +...+ (n+1)^(n-1)*x^n/n! +...;
W_2(x) = 1 + x + 5*x^2/2! + 43*x^3/3! + 525*x^4/4! + 8321*x^5/5! + 162463*x^6/6! + +...+ A227176(n)*x^n/n! +...;
W_3(x) = 1 + x + 7*x^2/2! + 82*x^3/3! + 1345*x^4/4! + 28396*x^5/5! + 734149*x^6/6! +...+ A268653(n)*x^n/n! +...;
W_4(x) = 1 + x + 9*x^2/2! + 133*x^3/3! + 2729*x^4/4! + 71721*x^5/5! + 2300485*x^6/6! +...+ A268654(n)*x^n/n! +...;
W_5(x) = 1 + x + 11*x^2/2! + 196*x^3/3! + 4821*x^4/4! + 151376*x^5/5! + 5787931*x^6/6! +...;
W_6(x) = 1 + x + 13*x^2/2! + 271*x^3/3! + 7765*x^4/4! + 283321*x^5/5! + 12567187*x^6/6! +...;
...
and satisfy:
(0) W_0(x) = exp(x),
(1) W_1(x) = exp(x)^W_1(x) = exp(T(x)) = LambertW(-x)/(-x),
(2) W_2(x) = W_1(x)^W_2(x) = exp(T(T(x))),
(3) W_3(x) = W_2(x)^W_3(x) = exp(T(T(T(x)))),
(4) W_4(x) = W_3(x)^W_4(x) = exp(T(T(T(T(x))))),
...
Euler's tree function T(x), and its iterates begin:
T(x) = x + 2*x^2/2! + 9*x^3/3! + 64*x^4/4! + 625*x^5/5! + 7776*x^6/6! + 117649*x^7/7! + 2097152*x^8/8! +...+ n^(n-1)*x^n/n! +...
T(T(x)) = x + 4*x^2/2! + 30*x^3/3! + 332*x^4/4! + 4880*x^5/5! + 89742*x^6/6! + 1986124*x^7/7! + 51471800*x^8/8! +...+ A207833(n)*x^n/n! +...
T(T(T(x))) = x + 6*x^2/2! + 63*x^3/3! + 948*x^4/4! + 18645*x^5/5! + 454158*x^6/6! + 13221075*x^7/7! + 448434136*x^8/8! +...+ A227278(n)*x^n/n! +...
T(T(T(T(x)))) = x + 8*x^2/2! + 108*x^3/3! + 2056*x^4/4! + 50680*x^5/5! + 1537524*x^6/6! + 55494712*x^7/7! + 2325685632*x^8/8! +...
...
Note that the e.g.f. of the n-th row function, W_n(x), also equals the ratio of two iterates of the Euler tree function: W_n(x) = T^n(x) / T^(n-1)(x).
See A274390 for the table of coefficients in these iterated tree functions.

Links

Crossrefs

Cf. A274390, A000272, A227176, A268653, A268654; diagonals: A274387, A274388.
Cf. A274741 (same table, but read differently).

Programs

  • PARI
    {ITERATE(F,n,k) = my(G=x +x*O(x^k)); for(i=1,n,G=subst(G,x,F));G}
{T(n,k) = my(TREE = serreverse(x*exp(-x +x*O(x^k)))); k!*polcoeff(exp(ITERATE(TREE,n,k)),k)}
/* Print this table as a square array */
for(n=0,10,for(k=0,10,print1(T(n,k),", "));print(""))
/* Print this table as a flattened array */
for(n=0,12,for(k=0,n,print1(T(n-k,k),", "));)

Formula

Let W_n(x) denote the e.g.f. of the n-th row function of this table, and T^n(x) the n-th iteration of Euler's tree function T(x) (cf. A274390), then
(1) W_n(x) = exp( T^n(x) ).
(2) W_n(x) = T^n(x) / T^(n-1)(x).
(3) W_n(x) = W_{n+1}( x/exp(x) ).
(4) W_n(x) = W_n( x/exp(x) )^W_n(x).

A268653 E.g.f.: exp( T(T(T(x))) ), where T(x) = -LambertW(-x) is Euler's tree function (A000169).

Original entry on oeis.org

1, 1, 7, 82, 1345, 28396, 734149, 22485898, 796769201, 32084546824, 1447917011461, 72411962077126, 3976481464087609, 237939307837951708, 15412492927027232261, 1074675869343994244266, 80270802348342665849569, 6395153963612453962942096, 541390375948749181692141061, 48536543026953818449535683054, 4594206854845500504888845269481, 457878082780635055560866092165156, 47930551834845432770784732668907205
Offset: 0

Views

Author

Paul D. Hanna, Feb 09 2016

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 7*x^2/2! + 82*x^3/3! + 1345*x^4/4! + 28396*x^5/5! + 734149*x^6/6! + 22485898*x^7/7! + 796769201*x^8/8! +...
where A(x) = A( x/exp(x) )^A(x).
RELATED SERIES.
Define W(x) = LambertW(-x)/(-x), where W(x) = exp(x*W(x)) and begins:
W(x) = 1 + x + 3*x^2/2! + 4^2*x^3/3! + 5^3*x^4/4! + 6^4*x^5/5! + 7^5*x^6/6! + 8^6*x^7/7! + 9^7*x^8/8! +...+ A000272(n+1)*x^n/n! +...
then
(1) A(x) = W( x*W(x) * W(x*W(x)) ),
(2) A(x) = W( x*W(x) )^A(x),
(3) A(x) = exp( A(x) * x*W(x) * W(x*W(x)) ),
(4) A(x/exp(x)) = W(x*W(x)).
Let G(x) = A(x/exp(x)), which begins:
G(x) = 1 + x + 5*x^2/2! + 43*x^3/3! + 525*x^4/4! + 8321*x^5/5! + 162463*x^6/6! + 3774513*x^7/7! + 101808185*x^8/8! +...+ A227176(n)*x^n/n! +...
then W(x), G(x), and A(x) are in the family of functions that begin:
(1) W(x) = exp(x)^W(x) = exp(T(x)),
(2) G(x) = W(x)^G(x) = exp(T(T(x))),
(3) A(x) = G(x)^A(x) = exp(T(T(T(x)))), ...
where T(x) = -LambertW(-x) is Euler's tree function:
T(x) = x + 2*x^2/2! + 3^2*x^3/3! + 4^3*x^4/4! + 5^4*x^5/5! + 6^5*x^6/! + 7^6*x^7/7! + 8^7*x^8/8! +...+ A000169(n)*x^n/n! +...

Links

Crossrefs

Cf. A227176, A227278, A000169, A000272.

Programs

  • PARI
    /* E.g.f.: A(x) = exp(T(T(T(x))) ) */
{a(n)=local(T=sum(k=1, n, k^(k-1)*x^k/k!)+x*O(x^n)); n!*polcoeff(exp(subst(T, x, subst(T, x, T))), n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* E.g.f.: A(x) = W( x*W(x) * W(x*W(x)) ) */
{a(n)=local(W=sum(k=0, n, (k+1)^(k-1)*x^k/k!)+x*O(x^n)); n!*polcoeff(subst(W, x, subst(x*W, x, x*W)), n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* E.g.f.: A(x) = exp( -A(x)*LambertW(LambertW(-x)) ) */
{a(n)=local(A=1+x, LambertW=sum(k=1, n, -k^(k-1)*(-x)^k/k!)+x*O(x^n));
for(i=1, n, A=exp(-A*subst(LambertW, x, subst(LambertW, x, -x)) +x*O(x^n))); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* E.g.f.: A(x) = ( LambertW(LambertW(-x))/LambertW(-x) )^A(x) */
{a(n)=local(A=1+x, W=sum(k=0, n, (k+1)^(k-1)*x^k/k!)+x*O(x^n));
for(i=1, n, A=subst(W,x,x*W)^A); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Formula

E.g.f. satisfies:
(1) A(x) = A(x/exp(x))^A(x).
(2) A(x) = W( x*W(x) * W(x*W(x)) ), where W(x) = LambertW(-x)/(-x).
(3) A(x) = W( x*W(x) )^A(x), where W(x) = LambertW(-x)/(-x).
(4) A(x) = exp( -A(x)*LambertW(LambertW(-x)) ).
(5) A(x) = ( LambertW(LambertW(-x)) / LambertW(-x) )^A(x).
(6) A(x/exp(x)) = exp(T(T(x))) = LambertW(LambertW(-x)) / LambertW(-x).
a(n) ~ exp(1 + (exp(-1) + exp(-1 - exp(-1)))*n) * n^(n-1) / sqrt((1 - exp(-1))*(1-exp(-1 - exp(-1)))). - Vaclav Kotesovec, Apr 01 2016

A268654 E.g.f.: exp( T(T(T(T(x)))) ), where T(x) = -LambertW(-x) is Euler's tree function (A000169).

Original entry on oeis.org

1, 1, 9, 133, 2729, 71721, 2300485, 87194689, 3815719969, 189440927857, 10525328121221, 647265172064985, 43660242639018241, 3205987437435132793, 254635755560090281525, 21755037223870035810001, 1989746853200670755116865, 194000891136578173746676449, 20089033883934411591428091013, 2202022786357483714102765694185
Offset: 0

Views

Author

Paul D. Hanna, Feb 18 2016

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 9*x^2/2! + 133*x^3/3! + 2729*x^4/4! + 71721*x^5/5! + 2300485*x^6/6! + 87194689*x^7/7! + 3815719969*x^8/8! +...
where A(x) = A( x/exp(x) )^A(x).
RELATED SERIES.
Define W(x) = LambertW(-x)/(-x), where W(x) = exp(x*W(x)) and begins:
W(x) = 1 + x + 3*x^2/2! + 4^2*x^3/3! + 5^3*x^4/4! + 6^4*x^5/5! + 7^5*x^6/6! + 8^6*x^7/7! + 9^7*x^8/8! +...+ A000272(n+1)*x^n/n! +...
Let F(x) = A(x/exp(x)), which begins:
F(x) = 1 + x + 5*x^2/2! + 43*x^3/3! + 525*x^4/4! + 8321*x^5/5! + 162463*x^6/6! + 3774513*x^7/7! + 101808185*x^8/8! +...+ A227176(n)*x^n/n! +...
Let G(x) = F(x/exp(x)), which begins:
G(x) = 1 + x + 7*x^2/2! + 82*x^3/3! + 1345*x^4/4! + 28396*x^5/5! + 734149*x^6/6! + 22485898*x^7/7! + 796769201*x^8/8! +...+ A268653(n)*x^n/n! +...
then W(x), F(x), G(x), and A(x) are in the family of functions that begin:
(1) W(x) = exp(x)^W(x) = exp(T(x)),
(2) F(x) = W(x)^F(x) = exp(T(T(x))),
(3) G(x) = F(x)^G(x) = exp(T(T(T(x)))),
(4) A(x) = G(x)^A(x) = exp(T(T(T(T(x))))), ...
where T(x) = -LambertW(-x) is Euler's tree function:
T(x) = x + 2*x^2/2! + 3^2*x^3/3! + 4^3*x^4/4! + 5^4*x^5/5! + 6^5*x^6/! + 7^6*x^7/7! + 8^7*x^8/8! +...+ A000169(n)*x^n/n! +...

Links

Crossrefs

Cf. A268653, A227176, A000272, A000169.

Programs

  • Mathematica
    CoefficientList[Series[E^(-ProductLog[ProductLog[ProductLog[ProductLog[-x]]]]), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Apr 01 2016 *)
  • PARI
    /* E.g.f.: A(x) = exp(T(T(T(T(x)))) ) */
{a(n)=local(T=sum(k=1, n, k^(k-1)*x^k/k!)+x*O(x^n)); n!*polcoeff(exp(subst(T, x, subst(T, x, subst(T, x, T)))), n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* E.g.f.: A(x) = exp( -A(x)*LambertW(LambertW(LambertW(-x))) ) */
{a(n)=local(A=1+x, LambertW=sum(k=1, n, -k^(k-1)*(-x)^k/k!)+x*O(x^n));
for(i=1, n, A=exp(-A*subst(LambertW, x, subst(LambertW, x, subst(LambertW, x,-x))) +x*O(x^n))); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Formula

E.g.f. satisfies:
(1) A(x) = A(x/exp(x))^A(x).
(2) A(x) = exp( A(x)*T(T(T(x))) ).
(3) A(x/exp(x)) = exp(T(T(T(x)))) = LambertW(LambertW(LambertW(-x))) / LambertW(LambertW(-x)).
a(n) ~ exp(1 + (exp(-1) + exp(-1 - exp(-1)) + exp(-1 - exp(-1) - exp(-1 - exp(-1))))*n) * n^(n-1) / sqrt((1 - exp(-1)) * (1 + LambertW(LambertW(-exp(-1 - exp(-1) - exp(-1 - exp(-1)) - exp(-1 - exp(-1) - exp(-1 - exp(-1))))))) * (1 + LambertW(-exp(-1 - exp(-1) - exp(-1 - exp(-1)) - exp(-1 - exp(-1) - exp(-1 - exp(-1))))))). - Vaclav Kotesovec, Apr 01 2016

A340473 a(n) = n! [x^n] W(-W(x))/(-W(x)), where W(x) is the Lambert W function.

Original entry on oeis.org

1, 1, 1, 7, 13, 321, 31, 42673, -214983, 12251809, -156239909, 6366130761, -135725103227, 5265915854785, -155145910919817, 6318044844152161, -232403136941014799, 10299509100942804033, -446889500139353805773, 21789892230658085847673, -1078684347590588362463619
Offset: 0

Views

Author

Peter Luschny, Jan 08 2021

Keywords

Comments

Let LW(x) = W(-W(x))/(-W(x)) denote the function in the definition and let T(x) = -W(-x) be Euler's tree function A000169, and L(x) = W(-x)/(-x) the labeled tree function A000272, then LW(x) = L(W(x)), and TW(x) := -T(W(-x)) is A097174, and RW(x) := T(-W(-x)) is A207833.

Links

Crossrefs

Cf. A097174, A177885, A207833, A227176, A340474.
Cf. A000169, A000272.

Programs

  • Maple
    W := x -> LambertW(x): gf := W(-W(x))/(-W(x)):
ser := series(gf, x, 24): seq(n!*coeff(ser, x, n), n=0..20);
  • Mathematica
    gf := -ProductLog[-ProductLog[x]]/ProductLog[x];
Range[0, 20]! CoefficientList[Series[gf, {x, 0, 20}], x]
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(lambertw(-lambertw(x))/(-lambertw(x)))) \\ Michel Marcus, Jan 09 2021

A340474 a(n) = n! [x^n] LW(T(x)), where T(x) = -W(-x) Euler's tree function, W(x) is the Lambert W function, and LW(x) = W(-W(x))/(-W(x)) (A340473).

Original entry on oeis.org

1, 1, 3, 22, 209, 2756, 43717, 839686, 18581425, 470707192, 13352676101, 420875581754, 14566375690297, 549877190829604, 22472783629465093, 989043215802778966, 46631075599107558113, 2345376059569552767344, 125350843842721213505029, 7095169059445749303612946
Offset: 0

Views

Author

Peter Luschny, Jan 09 2021

Keywords

Crossrefs

Cf. A340473, A097174, A177885, A207833, A227176.
Cf. A000169, A000272.

Programs

  • Maple
    W := x -> LambertW(x): T := x -> -W(-x): LW := x -> W(-W(x))/(-W(x)):
ser := series(LW(T(x)), x, 24): seq(n!*coeff(ser, x, n), n=0..19);
Showing 1-6 of 6 results.

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