A207833 -id:A207833 - 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) ).

A185298 Expansion of e.g.f. xexp(x)exp(x*exp(x)).

Original entry on oeis.org

0, 1, 4, 18, 92, 520, 3222, 21700, 157544, 1224576, 10133450, 88843084, 821832156, 7992373168, 81458868974, 867700216380, 9636146477648, 111323478770560, 1335253363581330, 16598183219157772, 213488758730421380, 2837046652845555696, 38899888173340835894

Offset: 0

Geoffrey Critzer, Feb 20 2011

nonn

a(n) is the number of ways to designate an element in each block of the set partitions of {1,2,...,n} and then designate a block.
Inverse binomial transform: b(n) = Sum (-1)^(n-k)*C(n,k)*a(k), k=0..n of A052512. - Alexander R. Povolotsky, Oct 01 2011
Number of pointed set partitions of pointed sets k[1...k...n] for any point k. - Gus Wiseman, Sep 27 2015
Exponential series reversal gives A207833 with alternating signs: 1, -4, 30, -332, 4880, ... . - Vladimir Reshetnikov, Aug 04 2019

			The a(2) = 4 pointed set partitions are 1[1[12]], 1[1[1]2[2]], 2[1[1]2[2]], 2[2[12]].
The a(3) = 18 pointed set partitions are 1[1[123]], 1[1[1]2[23]], 1[1[1]3[23]], 1[1[12]3[3]], 1[1[13]2[2]], 1[1[1]2[2]3[3]], 2[2[123]], 2[1[1]2[23]], 2[1[13]2[2]], 2[2[2]3[13]], 2[2[12]3[3]], 2[1[1]2[2]3[3]], 3[3[123]], 3[1[1]3[23]], 3[1[12]3[3]], 3[2[2]3[13]], 3[2[12]3[3]], 3[1[1]2[2]3[3]].

G. C. Greubel, Table of n, a(n) for n = 0..535
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Cf. A052512, A080108, A207833, A262671.

nn=30; a=x Exp[x]; Range[0,nn]! CoefficientList[Series[a Exp[a], {x,0,nn}],x]

x='x+O('x^33); concat([0],Vec(serlaplace(x*exp(x)*exp(x*exp(x))))) \\ Joerg Arndt, Oct 04 2015

E.g.f.: A(A(x)) where A(x) = x*exp(x).
a(n) = Sum_{k=1..n} binomial(n,k)*k^(n-k+1). - Vladimir Kruchinin, Sep 23 2011
O.g.f.: Sum_{k>=1} k*x^k/(1 - k*x)^(k+1). - Ilya Gutkovskiy, Oct 09 2018
a(n) ~ exp(r*exp(r) + r - n) * n^(n + 1/2) / (r^(n - 1/2) * sqrt(1 + exp(r)*(1 + 3*r + r^2))), where r = 2*LambertW(exp(1/4)*sqrt(n)/2) - 1/2 + 1/(16*LambertW(exp(1/4)*sqrt(n)/2)^2 + LambertW(exp(1/4)*sqrt(n)/2) - 1). - Vaclav Kotesovec, Mar 21 2023

A227176 E.g.f.: LambertW(LambertW(-x)) / LambertW(-x).

Original entry on oeis.org

1, 1, 5, 43, 525, 8321, 162463, 3774513, 101808185, 3129525793, 108063152091, 4143297446729, 174723134310277, 8039591465487297, 400924930695585143, 21543513647508536161, 1241094846565489688817, 76314967969651411780673, 4989260143610128556354611

Offset: 0

Paul D. Hanna, Jul 04 2013

nonn

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 43*x^3/3! + 525*x^4/4! + 8321*x^5/5! +...
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! +...
then
(1) A(x) = W(x*W(x)),
(4) A(x) = W(x)^A(x),
(3) A(x) = exp( x*A(x)*W(x) ),
(8) A(x/exp(x)) = W(x).
The e.g.f. also satisfies:
(6) A(x) = 1 + A(x)*x + A(x)*(2 + A(x))*x^2/2! + A(x)*(3 + A(x))^2*x^3/3! + A(x)*(4 + A(x))^3*x^4/4! + A(x)*(5 + A(x))^4*x^5/5! +...
and, for all real m,
(7) A(x)^m = 1 + m*A(x)*(1+m*A(x))^0*x^1/1! + m*A(x)*(2+m*A(x))^1*x^2/2! + m*A(x)*(3+m*A(x))^2*x^3/3! + m*A(x)*(4+m*A(x))^3*x^4/4! + m*A(x)*(5+m*A(x))^4*x^5/5! +...

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A207833, A195203, A000169.

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

{a(n) = if(n==0,1,sum(k=0,n,binomial(n,k)*k*(k+1)^(k-1)*n^(n-k-1)))}
for(n=0,20,print1(a(n),", "))

/* E.g.f.: A(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,x*W), n)}

/* E.g.f.: A(x) = exp(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,T)), n)}

/* E.g.f.: A(x) = exp( -A(x)*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,-x) +x*O(x^n)));n!*polcoeff(A, n)}

/* E.g.f.: A(x) = ( LambertW(-x)/(-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=W^A);n!*polcoeff(A, n)}

/* E.g.f.: A(x) = Sum_{n>=0} A(x)*(n + A(x))^(n-1) * x^n/n!. */
{a(n)=local(A=1+x);for(i=1,n,A=sum(k=0, n, A*(k+A)^(k-1)*x^k/k!)+x*O(x^n)); n!*polcoeff(A, n)}

a(n) = Sum_{k=0..n} binomial(n,k) * k*(k+1)^(k-1) * n^(n-k-1) for n>0 with a(0)=1.
E.g.f. A(x) satisfies:
(1) A(x) = W(x*W(x)), where W(x) = LambertW(-x)/(-x) = Sum_{n>=0} (n+1)^(n-1)*x^n/n!.
(2) A(x) = exp( T(T(x)) ), where T(x) = -LambertW(-x) is Euler's tree function (A000169).
(3) A(x) = exp( -A(x)*LambertW(-x) ).
(4) A(x) = ( LambertW(-x)/(-x) )^A(x).
(5) A(x) = ( Sum_{n>=0} (n+1)^(n-1)*x^n/n! )^A(x).
(6) A(x) = Sum_{n>=0} A(x)*(n + A(x))^(n-1) * x^n/n!.
(7) A(x)^m = Sum_{n>=0} m*A(x)*(n + m*A(x))^(n-1) * x^n/n!.
(8) A(x/exp(x)) = exp(T(x)) = LambertW(-x)/(-x).
(9) log(A(x)) = A(x) * Sum_{n>=1} n^(n-1) * x^n/n!, and equals the e.g.f. of A207833.
(10) A(x) = 1 + Sum_{n>=1} (n+1)^(n-1)*x^n/n! * Sum_{k>=0} n*(k+n)^(k-1)*x^k/k!.
a(n) ~ n! * (-exp((1+exp(-1))*n)/(sqrt(2*Pi*(1-exp(-1)))*n^(3/2) *LambertW(-exp(-1-exp(-1))))). - Vaclav Kotesovec, Jul 05 2013

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

Paul D. Hanna, Jun 19 2016

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.

			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.

Paul D. Hanna, Table of n, a(n) for n = 0..1080, of rows 0..45 of the flattened table.

Cf. A274390, A000272, A227176, A268653, A268654; diagonals: A274387, A274388.

Cf. A274741 (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(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),", "));)

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).

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

A290840 a(n) = n! * [x^n] exp(n*x)/(1 + LambertW(-x)).

Original entry on oeis.org

1, 2, 12, 117, 1584, 27525, 585108, 14726411, 428551616, 14161828185, 523952280900, 21456869976135, 963553844335536, 47078974421716757, 2486272976536821332, 141118622400977894475, 8566597074999702384384, 553816179165426157329201, 37985975117322654130568964

Offset: 0

Ilya Gutkovskiy, Aug 12 2017

nonn

G. C. Greubel, Table of n, a(n) for n = 0..365
N. J. A. Sloane, Transforms
Martin Svatoš, Peter Jung, Jan Tóth, Yuyi Wang, and Ondřej Kuželka, On Discovering Interesting Combinatorial Integer Sequences, arXiv:2302.04606 [cs.LO], 2023, p. 17.

Main diagonal of A290824.

Cf. A000312, A207833, A254382.

Table[n! * SeriesCoefficient[Exp[n*x]/(1 + LambertW[-x]), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 06 2017 *)

a(n) = A290824(n,n).
a(n) ~ exp(1/2 + n*exp(-1)) * n^n / sqrt(exp(1)-1). - Vaclav Kotesovec, Oct 06 2017
a(n) = Sum_{k=0..n} binomial(n,k)*n^(n-k)*k^k. - Fabian Pereyra, Jul 16 2024
E.g.f.: 1/((1+LambertW(-x))*(1+LambertW(LambertW(-x)))). - Fabian Pereyra, Jul 19 2024

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

Peter Luschny, Jan 08 2021

sign

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.

Eric Weisstein's World of Mathematics, Lambert W-Function.

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

Cf. A000169, A000272.

W := x -> LambertW(x): gf := W(-W(x))/(-W(x)):
ser := series(gf, x, 24): seq(n!*coeff(ser, x, n), n=0..20);

gf := -ProductLog[-ProductLog[x]]/ProductLog[x];
Range[0, 20]! CoefficientList[Series[gf, {x, 0, 20}], x]

my(x='x+O('x^25)); Vec(serlaplace(lambertw(-lambertw(x))/(-lambertw(x)))) \\ Michel Marcus, Jan 09 2021

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A185298 Expansion of e.g.f. x*exp(x)*exp(x*exp(x)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A227176 E.g.f.: LambertW(LambertW(-x)) / LambertW(-x).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

PARI

PARI

PARI

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A290840 a(n) = n! * [x^n] exp(n*x)/(1 + LambertW(-x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

A185298 Expansion of e.g.f. xexp(x)exp(x*exp(x)).