A000248 -id:A000248 - cp's OEIS frontend

A059297 Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 1.

1, 0, 1, 0, 2, 1, 0, 3, 6, 1, 0, 4, 24, 12, 1, 0, 5, 80, 90, 20, 1, 0, 6, 240, 540, 240, 30, 1, 0, 7, 672, 2835, 2240, 525, 42, 1, 0, 8, 1792, 13608, 17920, 7000, 1008, 56, 1, 0, 9, 4608, 61236, 129024, 78750, 18144, 1764, 72, 1, 0, 10, 11520, 262440

Offset: 0

N. J. A. Sloane, Jan 25 2001

T(n,k) = C(n,k)*k^(n-k) is the number of functions f from domain [n] to codomain [n+1] such that f(x)=n+1 for exactly k elements x of [n] and f(f(x))=n+1 for the remaining n-k elements x of [n]. Subsequently, row sums of T(n,k) provide the number of functions f:[n]->[n+1] such that either f(x)=n+1 or f(f(x))=n+1 for every x in [n]. We note that there are C(n,k) ways to choose the k elements mapped to n+1 and there are k^(n-k) ways to map n-k elements to a set of k elements. - Dennis P. Walsh, Sep 05 2012
Conjecture: the matrix inverse is A137452. - R. J. Mathar, Mar 12 2013
The above conjecture is correct. This triangle is the exponential Riordan array [1, x*exp(x)]. Thus the  inverse array is the exponential Riordan array [ 1, W(x)], which equals A137452. - Peter Bala, Apr 08 2013

			Triangle begins:
1;
0,  1;
0,  2,   1;
0,  3,   6,    1;
0,  4,  24,   12,    1;
0,  5,  80,   90,   20,   1;
0,  6, 240,  540,  240,  30,  1;
0,  7, 672, 2835, 2240, 525, 42,  1;
Row 4. Expansion of x^4 in terms of Abel polynomials:
x^4 = -4*x+24*x*(x+2)-12*x*(x+3)^2+x*(x+4)^3.
O.g.f. for column 2: A(-2,1/x) = x^2/(1-2*x)^3 = x^2+6*x^3+24*x^4+80*x^5+....

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 91, #43 and p. 135, [3i'].

Alois P. Heinz, Rows n = 0..140, flattened
Peter Bala, Diagonals of triangles with generating function exp(t*F(x)).
G. Duchamp, K. A. Penson, A. I. Solomon, A. Horzela and P. Blasiak, One-parameter groups and combinatorial physics, arXiv:quant-ph/0401126, 2004.
Emanuele Munarini, Combinatorial identities involving the central coefficients of a Sheffer matrix, Applicable Analysis and Discrete Mathematics (2019) Vol. 13, 495-517.
Bruce E. Sagan, A note on Abel polynomials and rooted labeled forests. Discrete Mathematics 44(3): 293-298 (1983).
J. Taylor, Formal group laws and hypergraph colorings, doctoral thesis, Univ. of Wash., 2016, p. 96, [_Tom Copeland_, Dec 20 2018].
Eric Weisstein's World of Mathematics, Abel Polynomial
Eric Weisstein's World of Mathematics, Idempotent Number
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.

There are 4 versions: A059297, A059298, A059299, A059300.
Diagonals give A001788, A036216, A040075, A050982, A002378, 3*A002417, etc.
Row sums are A000248.
Cf. A061356, A202017, A137452 (inverse array), A264428.

/* As triangle */ [[Binomial(n,k)*k^(n-k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Aug 22 2015

T:= (n, k)-> binomial(n, k) *k^(n-k):
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Sep 05 2012

nn=10;f[list_]:=Select[list,#>0&];Prepend[Map[Prepend[#,0]&,Rest[Map[f,Range[0,nn]!CoefficientList[Series[Exp[y x Exp[x]],{x,0,nn}],{x,y}]]]],{1}]//Grid  (* Geoffrey Critzer, Feb 09 2013 *)
t[n_, k_] := Binomial[n, k]*k^(n - k); Prepend[Flatten@Table[t[n, k], {n, 10}, {k, 0, n}], 1] (* Arkadiusz Wesolowski, Mar 23 2013 *)

# uses[bell_transform from A264428]
def A059297_row(n):
    nat = [k for k in (1..n)]
    return bell_transform(n, nat)
[A059297_row(n)  for n in range(8)] # Peter Luschny, Dec 20 2015

E.g.f.: exp(x*y*exp(y)). - Vladeta Jovovic, Nov 18 2003
Up to signs, this is the triangle of connection constants expressing the monomials x^n as a linear combination of the Abel polynomials A(k,x) := x*(x+k)^(k-1), 0 <= k <= n. O.g.f. for the k-th column: A(-k,1/x) = x^k/(1-k*x)^(k+1). Cf. A061356. Examples are given below. - Peter Bala, Oct 09 2011
The o.g.f.'s for the diagonals of this triangle are the rational functions occurring in the expansion of the compositional inverse (with respect to x) (x-t*x*exp(x))^-1 = x/(1-t) + 2*t/(1-t)^3*x^2/2! + (3*t+9*t^2)/(1-t)^5*x^3/3! + (4*t+52*t^2+64*t^3)/(1-t)^7*x^4/4! + .... For example, the o.g.f. for second subdiagonal is (3*t+9*t^2)/(1-t)^5 = 3*t + 24*t^2 + 90*t^3 + 240*t^4 + .... See the Bala link. The coefficients of the numerator polynomials are listed in A202017. - Peter Bala, Dec 08 2011
Recurrence equation: T(n+1,k+1) = Sum_{j=0..n-k} (j+1)*binomial(n,j)*T(n-j,k). - Peter Bala, Jan 13 2015
The Bell transform of [1,2,3,...]. See A264428 for the Bell transform. - Peter Luschny, Dec 20 2015

A087761 Expansion of (1-x)^(1/(x-1)).

Original entry on oeis.org

1, 1, 4, 21, 140, 1130, 10674, 115206, 1396016, 18739080, 275712840, 4408612560, 76070179272, 1408041937848, 27816773482848, 583970117197320, 12978149959718400, 304310928180279360, 7506092106055537344

Offset: 0

Vladeta Jovovic, Oct 02 2003

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..439

Cf. A000248, A005727.

Table[Sum[BellY[n, k, Table[m! HarmonicNumber[m], {m, n}]], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)

a(n)=if(n==0,1,(n-1)!*sum(k=0,n-1,(n-k)*sum(j=1,n-k,1/j)*a(k)/k!)) \\ Paul D. Hanna, Mar 17 2010; corrected Mar 19 2010

a(n) = Sum_{k=0..n} |Stirling1(n, k)|*A000248(k).
From Paul D. Hanna, Mar 17 2010: (Start)
E.g.f.: exp( Sum_{n>=1} H(n)*x^n ) where H(n) is the n-th harmonic number;
a(n) = (n-1)!*Sum_{k=0..n-1} (n-k)*H(n-k)*a(k)/k! for n>0 with a(0)=1. (End)
Empirical: a(n) = Sum_{i=0..n} binomial(n,i)*A005727(i)*(n-1)!/(i-1)! for n>0. - John M. Campbell, Dec 13 2016

A145460 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where sequence a_k of column k is the exponential transform of C(n,k).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 0, 3, 5, 1, 0, 1, 10, 15, 1, 0, 0, 3, 41, 52, 1, 0, 0, 1, 9, 196, 203, 1, 0, 0, 0, 4, 40, 1057, 877, 1, 0, 0, 0, 1, 10, 210, 6322, 4140, 1, 0, 0, 0, 0, 5, 30, 1176, 41393, 21147, 1, 0, 0, 0, 0, 1, 15, 175, 7273, 293608, 115975, 1, 0, 0, 0, 0, 0, 6, 35, 1176, 49932, 2237921, 678570

Offset: 0

Alois P. Heinz, Oct 10 2008

A(n,k) is also the number of ways of placing n labeled balls into indistinguishable boxes, where in each filled box k balls are seen at the top. E.g. A(3,1)=10:
  |1.| |2.| |3.| |1|2| |1|2| |1|3| |1|3| |2|3| |2|3| |1|2|3|
  |23| |13| |12| |3|.| |.|3| |2|.| |.|2| |1|.| |.|1| |.|.|.|
  +--+ +--+ +--+ +-+-+ +-+-+ +-+-+ +-+-+ +-+-+ +-+-+ +-+-+-+

			Square array A(n,k) begins:
   1,   1,  1,  1,  1,  1,  ...
   1,   1,  0,  0,  0,  0,  ...
   2,   3,  1,  0,  0,  0,  ...
   5,  10,  3,  1,  0,  0,  ...
  15,  41,  9,  4,  1,  0,  ...
  52, 196, 40, 10,  5,  1,  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
N. J. A. Sloane, Transforms

Columns k=0-9 give: A000110, A000248, A133189, A145453, A145454, A145455, A145456, A145457, A145458, A145459.
A(2n,n) gives A029651.
Cf.: A007318, A143398, A292948.

exptr:= proc(p) local g; g:=
          proc(n) option remember; `if`(n=0, 1,
             add(binomial(n-1, j-1) *p(j) *g(n-j), j=1..n))
        end: end:
A:= (n,k)-> exptr(i-> binomial(i, k))(n):
seq(seq(A(n, d-n), n=0..d), d=0..12);

Exptr[p_] := Module[{g}, g[n_] := g[n] = If[n == 0, 1, Sum[Binomial[n-1, j-1] *p[j]*g[n-j], {j, 1, n}]]; g]; A[n_, k_] := Exptr[Function[i, Binomial[i, k]]][n]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Jan 15 2014, translated from Maple *)

def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(k, n)
  ary = [1]
  (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + ncr(i - 1, j) * ncr(j + 1, k) * ary[i - 1 - j]}}
  ary
end
def A145460(n)
  a = []
  (0..n).each{|i| a << A(i, n - i)}
  ary = []
  (0..n).each{|i|
    (0..i).each{|j|
      ary << a[i - j][j]
    }
  }
  ary
end
p A145460(20) # Seiichi Manyama, Sep 28 2017

A(0,k) = 1 and A(n,k) = Sum_{i=0..n-1} binomial(n-1,i) * binomial(i+1,k) * A(n-1-i,k) for n > 0. - Seiichi Manyama, Sep 28 2017

A216857 Number of connected functions from {1,2,...,n} into a subset of {1,2,...,n} that have a fixed point summed over all subsets.

Original entry on oeis.org

0, 1, 4, 24, 224, 2880, 47232, 942592, 22171648, 600698880, 18422374400, 630897721344, 23864653578240, 988197253808128, 44460603225407488, 2159714024218951680, 112652924603290615808, 6280048587936003784704, 372616014329572403183616, 23445082059018189741752320, 1559275240299007139066675200

Offset: 0

Geoffrey Critzer, Sep 17 2012

nonn

Essentially the same as A038049.
Also the number of rooted trees whose nodes are labeled with the blocks of a set partition of {1,2,...,n} each having a distinguished element. (See A000248.)
The bijection is straightforward. The trees correspond to functional digraphs mapping the distinguished elements towards the root. All the elements within each block are mapped to the distinguished element of that block. The distinguished element in the root node is the fixed point.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.

Cf. A048802, A072034.

With[{nmax = 20}, CoefficientList[Series[-LambertW[-x*Exp[x]], {x, 0, nmax}], x]*Range[0, nmax]!] (* modified by G. C. Greubel, Nov 15 2017 *)

for(n=0,30, print1(sum(k=1,n, binomial(n,k)*k^(n-1)), ", ")) \\ G. C. Greubel, Nov 15 2017

my(x='x+O('x^50)); concat([0], Vec(serlaplace(-lambertw(-x*exp(x))))) \\ G. C. Greubel, Nov 15 2017

E.g.f.: T(x*exp(x)) where T(x) is the e.g.f. for A000169.
a(n) = Sum_{k=1..n} binomial(n,k)*k^(n-1).
a(n) ~ sqrt(1+LambertW(exp(-1))) * n^(n-1) / (exp(n)*LambertW(exp(-1))^n). - Vaclav Kotesovec, Jul 09 2013
O.g.f.: Sum_{n>=0} n^(n-1)* x^n / (1 - n*x)^(n+1). - Paul D. Hanna, May 22 2018
E.g.f.: the compositional inverse of A(x) is -A(-x). - Alexander Burstein, Aug 11 2018

A000951 Number of forests with n nodes and height at most 4.

Original entry on oeis.org

1, 3, 16, 125, 1296, 16087, 229384, 3687609, 66025360, 1303751051, 28151798544, 659841763957, 16681231615816, 452357366282655, 13095632549137576, 403040561722348913, 13138626717852194976, 452179922268565180819, 16381932383826669204640

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..100
John Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103.

Cf. A000248, A000949, A000950, A052512-A052514.

Column k=4 of A210725. - Alois P. Heinz, Mar 15 2013

nn = 20; Range[0, nn]! CoefficientList[Series[Exp[x*Exp[x*Exp[x*Exp[x*Exp[x]]]]], {x, 0, nn}], x] (* T. D. Noe, Jun 21 2012 *)

E.g.f.: exp(x*exp(x*exp(x*exp(x*exp(x))))).

More terms from Vladeta Jovovic, Apr 07 2001

A143398 Triangle T(n,k) = number of forests of labeled rooted trees of height at most 1, with n labels, where each root contains k labels, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 10, 3, 1, 0, 41, 9, 4, 1, 0, 196, 40, 10, 5, 1, 0, 1057, 210, 30, 15, 6, 1, 0, 6322, 1176, 175, 35, 21, 7, 1, 0, 41393, 7273, 1176, 105, 56, 28, 8, 1, 0, 293608, 49932, 7084, 756, 126, 84, 36, 9, 1, 0, 2237921, 372060, 42120, 6510, 378, 210, 120, 45, 10, 1

Offset: 0

Alois P. Heinz, Aug 12 2008

			T(4,2) = 9: 3->{1,2}<-4, 2->{1,3}<-4, 2->{1,4}<-3, 1->{2,3}<-4, 1->{2,4}<-3, 1->{3,4}<-2, {1,2}{3,4}, {1,3}{2,4}, {1,4}{2,3}.
Triangle begins:
  1;
  0,   1;
  0,   3,  1;
  0,  10,  3,  1;
  0,  41,  9,  4,  1;
  0, 196, 40, 10,  5,  1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Index entries for sequences related to rooted trees

Columns k=0-9 give: A000007, A000248, A133189, A145453, A145454, A145455, A145456, A145457, A145458, A145459.
Main diagonal gives A000012.
Row sums give A143406.
T(2n,n) gives A029651.
Cf. A000142, A145460, A351703.

u:= (n, k)-> `if`(k=0, 0, floor(n/k)):
T:= (n, k)-> n! *add(i^(n-k*i)/ ((n-k*i)! *i! *k!^i), i=0..u(n, k)):
seq(seq(T(n, k), k=0..n), n=0..12);

t[n_, n_] = 1; t[, 0] = 0; t[n, k_] := n!*Sum[i^(n-k*i)/((n-k*i)!*i!*k!^i), {i, 0, n/k}]; Table[t[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 19 2013 *)

u(n,k) = if(k==0, 0, n\k);
T(n, k) = n!*sum(j=0, u(n, k), j^(n-k*j)/(k!^j*j!*(n-k*j)!)); \\ Seiichi Manyama, May 13 2022

T(n,k) = n! * Sum_{i=0..u(n,k)} i^(n-k*i)/((n-k*i)!*i!*k!^i) with u(n,k) = 0 if k=0 and u(n,k) = floor(n/k) else.

A216507 E.g.f. exp( x^2 * exp(x) ).

Original entry on oeis.org

1, 0, 2, 6, 24, 140, 870, 5922, 45416, 381096, 3442410, 33382910, 345803172, 3801763836, 44156760830, 539962736250, 6929042527920, 93032248209872, 1303556965679826, 19018807375195638, 288341417011487420, 4534168069704168420, 73829219253218066022, 1242905562198878544626

Offset: 0

Joerg Arndt, Sep 14 2012

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..516 (terms 0..200 from Vincenzo Librandi)
Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function

Column k=2 of A292978.
Cf. A216688 (e.g.f. exp(x*exp(x^2))), A216689 (e.g.f. exp(x*exp(x)^2)).
Cf. A000248 (e.g.f. exp(x*exp(x))), A003725 (e.g.f. exp(x*exp(-x))).

With[{nn = 25}, CoefficientList[Series[Exp[x^2 Exp[x]], {x, 0, nn}],
   x] Range[0, nn]!] (* Bruno Berselli, Sep 14 2012 *)

x='x+O('x^66);
Vec(serlaplace(exp( x^2 * exp(x) )))
/* Joerg Arndt, Sep 14 2012 */

From Vaclav Kotesovec, Aug 06 2014: (Start)
a(n) ~ n^n / (exp(n*(1+r)/(2+r)) * r^n * sqrt((1+r)*(4+r)/(2+r))), where r is the root of the equation r^2*(2+r)*exp(r) = n.
(a(n)/n!)^(1/n) ~ exp(1/(3*LambertW(n^(1/3)/3))) / (3*LambertW(n^(1/3)/3)).
(End)
a(n) = Sum_{k = 0..n/2} C(n,2*k) * ((2*k)!/k!) * k^(n-2*k). - David Einstein, Oct 30 2016

A245501 Number A(n,k) of endofunctions f on [n] such that f^k(i) = f(i) for all i in [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 3, 27, 1, 1, 1, 4, 10, 256, 1, 1, 1, 3, 19, 41, 3125, 1, 1, 1, 4, 12, 110, 196, 46656, 1, 1, 1, 3, 19, 73, 751, 1057, 823543, 1, 1, 1, 4, 10, 116, 556, 5902, 6322, 16777216, 1, 1, 1, 3, 21, 41, 901, 4737, 52165, 41393, 387420489, 1

Offset: 0

Alois P. Heinz, Jul 24 2014

			Square array A(n,k) begins:
  1,     1,    1,    1,    1,    1,    1, ...
  1,     1,    1,    1,    1,    1,    1, ...
  1,     4,    3,    4,    3,    4,    3, ...
  1,    27,   10,   19,   12,   19,   10, ...
  1,   256,   41,  110,   73,  116,   41, ...
  1,  3125,  196,  751,  556,  901,  220, ...
  1, 46656, 1057, 5902, 4737, 8422, 1921, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Column k=0-10 give: A000012, A000312, A000248, A060905, A060906, A060907, A245502, A245503, A245504, A245505, A245506.

Main diagonal gives A245507.

with(numtheory):
A:= (n, k)-> `if`(k=0, 1, `if`(k=1, n^n, n! *coeff(series(
    exp(add((x*exp(x))^d/d, d=divisors(k-1))), x, n+1), x, n))):
seq(seq(A(n, d-n), n=0..d), d=0..12);

A[0, 1] = 1; A[n_, k_] := If[k==0, 1, If[k==1, n^n, n!*SeriesCoefficient[ Exp[ DivisorSum[k-1, (x*Exp[x])^#/#&]], {x, 0, n}]]]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Mar 20 2017, translated from Maple *)

A(n,k) = n! * [x^n] exp(Sum_{d|(k-1)} (x*exp(x))^d/d) for k>1, A(n,0)=1, A(n,1)=n^n.

A279636 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the exponential transform of the k-th powers.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 5, 10, 15, 1, 1, 9, 22, 41, 52, 1, 1, 17, 52, 125, 196, 203, 1, 1, 33, 130, 413, 836, 1057, 877, 1, 1, 65, 340, 1445, 3916, 6277, 6322, 4140, 1, 1, 129, 922, 5261, 19676, 41077, 52396, 41393, 21147, 1, 1, 257, 2572, 19685, 104116, 288517, 481384, 479593, 293608, 115975

Offset: 0

Alois P. Heinz, Dec 16 2016

			Square array A(n,k) begins:
:   1,    1,    1,     1,      1,       1,        1, ...
:   1,    1,    1,     1,      1,       1,        1, ...
:   2,    3,    5,     9,     17,      33,       65, ...
:   5,   10,   22,    52,    130,     340,      922, ...
:  15,   41,  125,   413,   1445,    5261,    19685, ...
:  52,  196,  836,  3916,  19676,  104116,   572036, ...
: 203, 1057, 6277, 41077, 288517, 2133397, 16379797, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Wikipedia, Kronecker delta

Columns k=0-10 give: A000110, A000248, A033462, A279358, A279637, A279638, A279639, A279640, A279641, A279642, A279643.
Rows n=0+1,2 give: A000012, A000051.
Main diagonal gives A279644.
Cf. A145460.

egf:= k-> exp(exp(x)*add(Stirling2(k, j)*x^j, j=0..k)-`if`(k=0, 1, 0)):
A:= (n, k)-> n!*coeff(series(egf(k), x, n+1), x, n):
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
A:= proc(n, k) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1)*j^k*A(n-j, k), j=1..n))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

A[n_, k_] := A[n, k] = If[n==0, 1, Sum[Binomial[n-1, j-1]*j^k*A[n-j, k], {j, 1, n}]]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 19 2017, translated from Maple *)

E.g.f. of column k: exp(exp(x)*(Sum_{j=0..k} Stirling2(n,j)*x^j) - delta_{0,k}).

A216689 Expansion of e.g.f. exp( x * exp(x)^2 ).

Original entry on oeis.org

1, 1, 5, 25, 153, 1121, 9373, 87417, 898033, 10052353, 121492341, 1573957529, 21729801481, 318121178337, 4917743697805, 79981695655801, 1364227940101857, 24335561350365953, 452874096174214117, 8772713803852981785, 176541611843378273401, 3684142819311127955041, 79596388271096140589949

Offset: 0

Joerg Arndt, Sep 14 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function

Cf. A216507 (e.g.f. exp(x^2*exp(x))), A216688 (e.g.f. exp(x*exp(x^2))).
Cf. A000248 (e.g.f. exp(x*exp(x))), A003725 (e.g.f. exp(x*exp(-x))).
Cf. A240165 (e.g.f. exp(x*(1+exp(x)^2))).

With[{nn = 25}, CoefficientList[Series[Exp[x Exp[x]^2], {x, 0, nn}], x] Range[0, nn]!] (* Bruno Berselli, Sep 14 2012 *)

x='x+O('x^66);
Vec(serlaplace(exp( x * exp(x)^2 )))
/* Joerg Arndt, Sep 14 2012 */

/* From o.g.f.: */
{a(n)=local(A=1);A=sum(k=0, n, x^k/(1 - 2*k*x +x*O(x^n))^(k+1));polcoeff(A, n)}
for(n=0,25,print1(a(n),", ")) /* Paul D. Hanna, Aug 02 2014 */

/* From binomial sum: */
{a(n)=sum(k=0,n, binomial(n,k)*(2*k)^(n-k))}
for(n=0,30,print1(a(n),", ")) /* Paul D. Hanna, Aug 02 2014 */

O.g.f.: Sum_{n>=0} x^n / (1 - 2*n*x)^(n+1). - Paul D. Hanna, Aug 02 2014
a(n) = Sum_{k=0..n} binomial(n,k) * (2*k)^(n-k) for n>=0. - Paul D. Hanna, Aug 02 2014
From Vaclav Kotesovec, Aug 06 2014: (Start)
a(n) ~ n^n / (exp(2*n*r/(1+2*r)) * r^n * sqrt((1+6*r+4*r^2)/(1+2*r))), where r is the root of the equation r*(1+2*r)*exp(2*r) = n.
(a(n)/n!)^(1/n) ~ exp(1/(2*LambertW(sqrt(n/2)))) / LambertW(sqrt(n/2)).
(End)

cp's OEIS Frontend

A059297 Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 1.

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A145460 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where sequence a_k of column k is the exponential transform of C(n,k).

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A216857 Number of connected functions from {1,2,...,n} into a subset of {1,2,...,n} that have a fixed point summed over all subsets.

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A000951 Number of forests with n nodes and height at most 4.

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A143398 Triangle T(n,k) = number of forests of labeled rooted trees of height at most 1, with n labels, where each root contains k labels, n>=0, 0<=k<=n.

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A216507 E.g.f. exp( x^2 * exp(x) ).

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