A000169 -id:A000169 - cp's OEIS frontend

A274390 Table of coefficients in the iterations of Euler's tree function (A000169), as read by antidiagonals.

1, 1, 0, 1, 2, 0, 1, 4, 9, 0, 1, 6, 30, 64, 0, 1, 8, 63, 332, 625, 0, 1, 10, 108, 948, 4880, 7776, 0, 1, 12, 165, 2056, 18645, 89742, 117649, 0, 1, 14, 234, 3800, 50680, 454158, 1986124, 2097152, 0, 1, 16, 315, 6324, 112625, 1537524, 13221075, 51471800, 43046721, 0, 1, 18, 408, 9772, 219000, 4090980, 55494712, 448434136, 1530489744, 1000000000, 0, 1, 20, 513, 14288, 387205, 9266706, 176238685, 2325685632, 17386204761, 51395228090, 25937424601, 0, 1, 22, 630, 20016, 637520, 18704322, 463975764, 8793850560, 111107380464, 759123121050, 1924687118684, 743008370688, 0, 1, 24, 759, 27100, 993105, 34617288, 1067280319, 26858490392, 499217336145, 5964692819140, 36882981687519, 79553145323940, 23298085122481, 0

Offset: 0

Paul D. Hanna, Jun 19 2016

See table A274391 for the coefficients in exp( T^n(x) ), n>=0, where T^n(x) is the e.g.f. of the n-th row of this table.

			This table begins:
1,  0,   0,     0,       0,        0,          0,            0, ...;
1,  2,   9,    64,     625,     7776,     117649,      2097152, ...;
1,  4,  30,   332,    4880,    89742,    1986124,     51471800, ...;
1,  6,  63,   948,   18645,   454158,   13221075,    448434136, ...;
1,  8, 108,  2056,   50680,  1537524,   55494712,   2325685632, ...;
1, 10, 165,  3800,  112625,  4090980,  176238685,   8793850560, ...;
1, 12, 234,  6324,  219000,  9266706,  463975764,  26858490392, ...;
1, 14, 315,  9772,  387205, 18704322, 1067280319,  70311813880, ...;
1, 16, 408, 14288,  637520, 34617288, 2217367600, 163802295616, ...;
1, 18, 513, 20016,  993105, 59879304, 4254311817, 348285415872, ...;
1, 20, 630, 27100, 1480000, 98110710, 7656893020, 688058734520, ...;
...
where the e.g.f.s of the rows are iterations of T(x) and begin:
T^0(x) = x;
T^1(x) = 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^2(x) = 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^3(x) = 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^4(x) = 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! +...;
...
where T^n(x)/exp( T^n(x) ) = T^n( x/exp(x) ) = T^(n-1)(x).
Also we have
T(x) = x*exp( T(x) );
T^2(x) = x*exp( T(x) + T^2(x) );
T^3(x) = x*exp( T(x) + T^2(x) + T^3(x) );
T^4(x) = x*exp( T(x) + T^2(x) + T^3(x) + T^4(x) ); ...

Paul D. Hanna, Table of n, a(n) for n = 0..1034 of rows 0..45 of the flattened table.

Cf. A274391, A000169, A207833, A227278; diagonals: A274389, A274392.
Cf. A274570 (transforms diagonals).
Cf. A274740 (same table, but read differently).

{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(ITERATE(TREE,n,k),k)}
/* Print this table as a square array */
for(n=0,10,for(k=1,10,print1(T(n,k),", "));print(""))
/* Print this table as a flattened array */
for(n=0,12,for(k=1,n,print1(T(n-k,k),", "));)

Let T^n(x) denote the n-th iteration of Euler's tree function T(x), then the coefficients in T^n(x) form the n-th row of this table, and the functions satisfy:
(1) T^n(x) = x * exp( Sum_{i=1..n} T^i(x) ).
(2) T^n(x) = T^(n-1)(x) * exp( T^n(x) ).
(3) T^n(x) = T^(n+1)( x/exp(x) ).

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

N. J. A. Sloane, Feb 20 2012

nonn

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

			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)).

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Yoshida Tomoyuki, Categorical aspects of generating functions. I. Exponential formulas and Krull-Schmidt categories, J. Algebra 240 (2001), no. 1, 40-82. MR1830543 (2002e:18008). See Sect. 6.8.

Cf. A000169, A185298, A227278, A227176.

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 *)

{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

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

A055860 a(n) = A000169(n+1) if n > 0; a(0) = 0.

Original entry on oeis.org

0, 2, 9, 64, 625, 7776, 117649, 2097152, 43046721, 1000000000, 25937424601, 743008370688, 23298085122481, 793714773254144, 29192926025390625, 1152921504606846976, 48661191875666868481, 2185911559738696531968

Offset: 0

Wolfdieter Lang, Jun 20 2000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 67

Second column of triangle A055858. Cf. A000169, A055858, A000312.

a(0) = 0; for n >= 1, a(n) = (n+1)^n.

E.g.f.: -W(-x)/((1+W(-x))*x) - 1 = -(d/dx)W(x) - 1, W(x) principal branch of Lambert's function.

A274570 Triangle, read by rows, that transforms diagonals in the array A274390 of coefficients in successive iterations of Euler's tree function (A000169).

Original entry on oeis.org

1, 1, 1, 7, 2, 1, 127, 20, 3, 1, 4377, 470, 39, 4, 1, 245481, 19912, 1125, 64, 5, 1, 20391523, 1326382, 56505, 2188, 95, 6, 1, 2354116899, 127677580, 4354923, 127056, 3755, 132, 7, 1, 360734454993, 16767030632, 476265591, 11117244, 247465, 5922, 175, 8, 1, 70865037282673, 2880746218304, 70056231213, 1360983976, 24228925, 436632, 8785, 224, 9, 1, 17367953099244051, 627213971899610, 13329387478113, 221585119536, 3281909155, 47290506, 716457, 12440, 279, 10, 1

Offset: 0

Paul D. Hanna, Jun 28 2016

This triangle also transforms diagonals in the array A274391 into each other, if we omit column 0 from those diagonals. The e.g.f. of row n of array A274391 equals exp(T^n(x)), where T^n(x) denotes the n-th iteration of Euler's tree function (A000169).

			This triangle T(n,k), n>=0, k=0..n, begins:
  1;
  1, 1;
  7, 2, 1;
  127, 20, 3, 1;
  4377, 470, 39, 4, 1;
  245481, 19912, 1125, 64, 5, 1;
  20391523, 1326382, 56505, 2188, 95, 6, 1;
  2354116899, 127677580, 4354923, 127056, 3755, 132, 7, 1;
  360734454993, 16767030632, 476265591, 11117244, 247465, 5922, 175, 8, 1;
  70865037282673, 2880746218304, 70056231213, 1360983976, 24228925, 436632, 8785, 224, 9, 1;
  17367953099244051, 627213971899610, 13329387478113, 221585119536, 3281909155, 47290506, 716457, 12440, 279, 10, 1;
  ...
Let D denote the triangular matrix defined by D(n,k) = T(n,k)/(n-k)!, such that D begins:
  1;
  1, 1;
  7/2!, 2, 1;
  127/3!, 20/2!, 3, 1;
  4377/4!, 470/3!, 39/2!, 4, 1;
  245481/5!, 19912/4!, 1125/3!, 64/2!, 5, 1;
  20391523/6!, 1326382/5!, 56505/4!, 2188/3!, 95/2!, 6, 1;
  ...
then D transforms diagonals in the array A274390 into each other:
  D * [1, 2/2, 30/3!, 948/4!, 50680/5!, 4090980/6!, ...] =
  [1, 4/2!, 63/3!, 2056/4!, 112625/5!, 9266706/6!, ...];
  D * [1, 4/2!, 63/3!, 2056/4!, 112625/5!, 9266706/6!, ...] =
  [1, 6/2!, 108/3!, 3800/4!, 219000/5!, 18704322/6!, ...];
  D * [1, 6/2!, 108/3!, 3800/4!, 219000/5!, 18704322/6!, ...] =
  [1, 8/2!, 165/3!, 6324/4!, 387205/5!, 34617288/6!, ...];
  ...
where array A274390 consists of coefficients in the iterations of Euler's tree function (A000169), and begins:
  1,  0,   0,     0,       0,        0,          0, ...;
  1,  2,   9,    64,     625,     7776,     117649, ...;
  1,  4,  30,   332,    4880,    89742,    1986124, ...;
  1,  6,  63,   948,   18645,   454158,   13221075, ...;
  1,  8, 108,  2056,   50680,  1537524,   55494712, ...;
  1, 10, 165,  3800,  112625,  4090980,  176238685, ...;
  1, 12, 234,  6324,  219000,  9266706,  463975764, ...;
  1, 14, 315,  9772,  387205, 18704322, 1067280319, ...;
  1, 16, 408, 14288,  637520, 34617288, 2217367600, ...;
  ...
Note that this triangle also transforms the diagonals of table A274391 into each other, if we omit column 0 from those diagonals.
After truncating column 0, table A274391 begins:
  1,  1,   1,     1,       1,         1,          1, ...;
  1,  3,  16,   125,    1296,     16807,     262144, ...;
  1,  5,  43,   525,    8321,    162463,    3774513, ...;
  1,  7,  82,  1345,   28396,    734149,   22485898, ...;
  1,  9, 133,  2729,   71721,   2300485,   87194689, ...;
  1, 11, 196,  4821,  151376,   5787931,  261066156, ...;
  1, 13, 271,  7765,  283321,  12567187,  656778529, ...;
  1, 15, 358, 11705,  486396,  24539593, 1457297878, ...;
  ...
for which the e.g.f. of row n equals exp(T^n(x)) - 1, where T^n(x) denotes the n-th iteration of Euler's tree function (A000169).
For example:
  D * [1, 3/2!, 43/3!, 1345/4!, 71721/5!, 5787931/6!, ...] =
  [1, 5/2!, 82/3!, 2729/4!, 151376/5!, 12567187/6!, ...];
  D * [1, 5/2!, 82/3!, 2729/4!, 151376/5!, 12567187/6!, ...] =
  [1, 7/2!, 133/3!, 4821/4!, 283321/5!, 24539593/6!, ...];
  D * [1, 7/2!, 133/3!, 4821/4!, 283321/5!, 24539593/6!, ...] =
  [1, 9/2!, 196/3!, 7765/4!, 486396/5!, 44223529/6!, ...];
  ...
The matrix inverse of triangle D, as shown with elements [D^-1][n,k] * (n-k)!, begins:
  1;
  -1, 1;
  -3, -2, 1;
  -40, -8, -3, 1;
  -1155, -140, -15, -4, 1;
  -57696, -5040, -324, -24, -5, 1;
  -4417175, -302092, -13923, -616, -35, -6, 1;
  -479964528, -26990720, -970848, -30720, -1040, -48, -7, 1;
  -70186001319, -3352727646, -98952435, -2439864, -58995, -1620, -63, -8, 1;
  -13284014648320, -551688200000, -13810202640, -279099200, -5254000, -102960, -2380, -80, -9, 1;
  -3158467118697099, -116039984093000, -2522473482375, -43202840076, -666167975, -10157796, -167475, -3344, -99, -10, 1;
  ...
The matrix square of triangle D, as shown with elements [D^2][n,k] * (n-k)!, begins:
  1;
  2, 1;
  18, 4, 1;
  377, 52, 6, 1;
  14304, 1414, 102, 8, 1;
  859977, 65904, 3411, 168, 10, 1;
  75306424, 4699274, 188496, 6668, 250, 12, 1;
  9061819643, 476161840, 15542811, 426144, 11485, 348, 14, 1;
  1435831150784, 65093379838, 1788015528, 39885108, 833280, 18162, 462, 16, 1;
  289948340816657, 11551390491440, 273593165397, 5134299808, 87266525, 1474704, 26999, 592, 18, 1;
  ...

Paul D. Hanna, Table of n, a(n) for n = 0..860, of rows 0..40 of flattened triangle.

Cf. A274390, A274571, A274572, A274573, A274574.

{T(n, k)=local(F=x,
LW=serreverse(x*exp(-x+x*O(x^(n+2)))), M, N, P, m=max(n, k));
M=matrix(m+3, m+3, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, LW)); polcoeff(F, c));
N=matrix(m+1, m+1, r, c, M[r, c]);
P=matrix(m+1, m+1, r, c, M[r+1, c]);
(n-k)!*(P~*N~^-1)[n+1, k+1]}
/* Print this triangle: */
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

A227278 E.g.f.: T(T(T(x))), where T(x) = -LambertW(-x) is Euler's tree function (A000169).

Original entry on oeis.org

1, 6, 63, 948, 18645, 454158, 13221075, 448434136, 17386204761, 759123121050, 36882981687519, 1974616464026484, 115536647641839333, 7336947898087080406, 502660682907018997755, 36961205206337621142192, 2903732354672613314658225, 242753209611983811853905330

Offset: 1

Paul D. Hanna, Jul 04 2013

nonn

			E.g.f.: A(x) = x + 6*x^2/2! + 63*x^3/3! + 948*x^4/4! + 18645*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! +...
where A(x) = T(T(T(x))).
Related expansions:
A(x/exp(x)) = A(x)/exp(A(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! +...
exp(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! +...+ A268653(n)*x^n/n! +...

Alois P. Heinz, Table of n, a(n) for n = 1..100

Cf. A268653, A207833, A000169.

Rest[CoefficientList[Series[-LambertW[LambertW[LambertW[-x]]], {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Jul 05 2013 *)

/* E.g.f.: A(x) = T(T(T(x))) */
{a(n)=local(T=sum(k=1, n, k^(k-1)*x^k/k!)+x*O(x^n)); n!*polcoeff(subst(T,x,subst(T, x, T)), n)}
for(n=1, 20, print1(a(n), ", "))

/* E.g.f.: A(x) = -LambertW(LambertW(LambertW(-x))) */
{a(n)=local(LambertW=sum(k=1, n, -k^(k-1)*(-x)^k/k!)+x*O(x^n));
n!*polcoeff(-subst(LambertW,x,subst(LambertW,x,subst(LambertW,x,-x))),n)}
for(n=1, 20, print1(a(n), ", "))

Given e.g.f. A(x), A(x/exp(x)) = A(x)/exp(A(x)) = T(T(x)) and equals the e.g.f. of A207833.

a(n) ~ n! * exp((1+exp(-1)+exp(-1-exp(-1)))*n)/(sqrt(2*Pi*(1-exp(-1))*(1-exp(-1-exp(-1))))*n^(3/2)). - Vaclav Kotesovec, Jul 05 2013

A383985 Series expansion of the exponential generating function LambertW(1-exp(x)), see A000169.

Original entry on oeis.org

0, 1, -1, 4, -23, 181, -1812, 22037, -315569, 5201602, -97009833, 2019669961, -46432870222, 1168383075471, -31939474693297, 942565598033196, -29866348653695203, 1011335905644178273, -36446897413531401020, 1392821757824071815641, -56259101478392975833333

Offset: 0

Michael De Vlieger, May 16 2025

Michael De Vlieger, Table of n, a(n) for n = 0..200
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 28, Table 2, permutative commutative operad "Perm/Com2".

Cf. A002050, A006531, A084099, A101851, A114285, A177885, A225883, A383986, A383987, A383988, A383989.

Composition of A000169 with signs and 1-exp(x).

nn = 20; f[x_] := -Sum[k^(k - 1)*(1 - Exp[x])^k/k!, {k, nn}];
Range[0, nn]! * CoefficientList[Series[f[x], {x, 0, nn}], x]

A038051 G.f.: B(x/(1-x)) where B is g.f. of A000169.

Original entry on oeis.org

1, 3, 14, 98, 944, 11642, 175108, 3108310, 63601168, 1473864722, 38152990484, 1091172974102, 34169139856024, 1162736848398010, 42723615842296540, 1685853467536076798, 71101435046807892512, 3191843270961299033762, 151956292916451992949028

Offset: 1

Christian G. Bower, Jan 04 1999

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A036249, A038052.

E.g.f. of A048802.

CoefficientList[Series[E^x*(-LambertW[-x]/(1+LambertW[-x])/x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Feb 17 2014 *)

E.g.f.: int(exp(x)*(-LambertW(-x)/(1+LambertW(-x))/x), x). a(n) = Sum_{k=0..n-1} binomial(n-1, k)*(k+1)^k. - Vladeta Jovovic, Apr 12 2003

a(n) ~ n^(n-1) * exp(exp(-1)). - Vaclav Kotesovec, Feb 17 2014

Corrected by Christian G. Bower, Mar 15 1999

A152917 A000169 prefixed by an initial 0.

Original entry on oeis.org

0, 1, 2, 9, 64, 625, 7776, 117649, 2097152, 43046721, 1000000000, 25937424601, 743008370688, 23298085122481, 793714773254144, 29192926025390625, 1152921504606846976, 48661191875666868481, 2185911559738696531968, 104127350297911241532841

Offset: 0

ShaoJun Ying (dolphinysj(AT)gmail.com), Dec 15 2008

nonn

A variant of A000169, which is the main entry for this sequence. - N. J. A. Sloane, Dec 19 2008

			a(10) = 10^9 = 1000000000.

Cf. A000169.

Join[{0},Table[n^(n-1),{n,20}]] (* Harvey P. Dale, Jan 28 2017 *)

a(n) = 0 if n = 0, otherwise a(n) = n^(n-1).
E.g.f.: A(x)=x*G(0) ; G(k)= 1 + x*(2*k+2)^(2*k)/((2*k+1)^(2*k) - x*(2*k+1)^(2*k)*(2*k+3)^(2*k+1)/(x*(2*k+3)^(2*k+1) + (2*k+2)^(2*k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Dec 30 2011
E.g.f.: -LambertW(-x). - Alois P. Heinz, Feb 26 2020

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

Paul D. Hanna, Feb 09 2016

nonn

			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! +...

Alois P. Heinz, Table of n, a(n) for n = 0..150

Cf. A227176, A227278, A000169, A000272.

/* 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), ", "))

/* 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), ", "))

/* 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), ", "))

/* 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), ", "))

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

A274389 Main diagonal of rectangular array A274390 of coefficients in the iterations of Euler's tree function (A000169).

Original entry on oeis.org

1, 2, 30, 948, 50680, 4090980, 463975764, 70311813880, 13718193268896, 3348658563980040, 999698412743754460, 358297471515195652308, 151813934699349280088328, 75064081768759279536110316, 42833194538353991390132088540, 27937122503026656234469859408880, 20653210428143999114034181337343616, 17178393944175652034128269331788145680, 15970217696130529428248774113884778921452, 16497536217367322285994072192399435877530380, 18836957575278690757486149667782477659475272520

Offset: 0

Paul D. Hanna, Jun 24 2016

nonn

Paul D. Hanna, Table of n, a(n) for n = 0..100

Cf. A274390, A000169, A274392.

{ITERATE(F,n,k) = my(G=x +x*O(x^k)); for(i=1,n,G=subst(G,x,F));G}
{A274390(n,k) = my(TREE = serreverse(x*exp(-x +x*O(x^k)))); k!*polcoeff(ITERATE(TREE,n,k),k)}
/* Print A274390 */
for(n=0,10,for(k=1,10,print1(A274390(n,k),", "));print("..."))
/* Print this sequence, as the main diagonal of A274390 */
for(n=0,20,print1(A274390(n,n+1),", "))

cp's OEIS Frontend

A274390 Table of coefficients in the iterations of Euler's tree function (A000169), as read by antidiagonals.

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

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A055860 a(n) = A000169(n+1) if n > 0; a(0) = 0.

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A274570 Triangle, read by rows, that transforms diagonals in the array A274390 of coefficients in successive iterations of Euler's tree function (A000169).

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A227278 E.g.f.: T(T(T(x))), where T(x) = -LambertW(-x) is Euler's tree function (A000169).

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A383985 Series expansion of the exponential generating function LambertW(1-exp(x)), see A000169.

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A038051 G.f.: B(x/(1-x)) where B is g.f. of A000169.

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A152917 A000169 prefixed by an initial 0.

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