A227278 -id:A227278 - 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

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).

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

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

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

PARI

PARI

PARI

Formula