A059297 -id:A059297 - cp's OEIS frontend

A059298 Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 2.

1, 2, 1, 3, 6, 1, 4, 24, 12, 1, 5, 80, 90, 20, 1, 6, 240, 540, 240, 30, 1, 7, 672, 2835, 2240, 525, 42, 1, 8, 1792, 13608, 17920, 7000, 1008, 56, 1, 9, 4608, 61236, 129024, 78750, 18144, 1764, 72, 1, 10, 11520, 262440, 860160, 787500, 272160, 41160

Offset: 0

N. J. A. Sloane, Jan 25 2001

The inverse triangle is the signed version 1,-2,1,9,-6,1,.. of triangle A061356. - Peter Luschny, Mar 13 2009
T(n,k) is the sum of the products of the cardinality of the blocks (cells) in the set partitions of {1,2,..,n} into exactly k blocks.
From Peter Bala, Jul 22 2014: (Start)
Exponential Riordan array [(1+x)*exp(x), x*exp(x)].
Let M = A093375, the exponential Riordan array [(1+x)*exp(x), x], and for k = 0,1,2,... define M(k) to be the lower unit triangular block array
  /I_k 0\
  \ 0  M/
having the k x k identity matrix I_k as the upper left block; in particular, M(0) = M. The present triangle equals the infinite matrix product M(0)*M(1)*M(2)*... - see the Example section. (End)
The Bell transform of n+1. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

			Triangle begins
1;
2, 1;
3, 6, 1;
4, 24, 12, 1; ...
From _Peter Bala_, Jul 22 2014: (Start)
With the arrays M(k) as defined in the Comments section, the infinite product M(0)*M(1)*M(2)*... begins
/1          \/1        \/1        \      /1           \
|2  1       ||0 1      ||0 1      |      |2  1        |
|3  4  1    ||0 2 1    ||0 0 1    |... = |3  6  1     |
|4  9  6 1  ||0 3 4 1  ||0 0 2 1  |      |4 24 12  1  |
|5 16 18 8 1||0 4 9 6 1||0 0 3 4 1|      |5 80 90 20 1|
|...        ||...      ||...      |      |...         | (End)

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

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Eric Weisstein's World of Mathematics, Idempotent Number

There are 4 versions: A059297, A059298, A059299, A059300.
Diagonals give A001788, A036216, A040075, A050982, A002378, 3*A002417, etc.
Row sums are A000248. A093375.

/* As triangle */ [[Binomial(n,k)*k^(n-k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Aug 22 2015

T:= (n, k)-> binomial(n+1,k+1)*(k+1)^(n-k): seq(seq(T(n, k), k=0..n), n=0..10); # Georg Fischer, Oct 27 2021

t = Transpose[ Table[ Range[0, 11]! CoefficientList[ Series[(x Exp[x])^n/n!, {x, 0, 11}], x], {n, 11}]]; Table[ t[[n, k]], {n, 2, 11}, {k, n - 1}] // Flatten (* or simply *)
t[n_, k_] := Binomial[n, k]*k^(n - k); Table[t[n, k], {n, 10}, {k, n}] // Flatten

for(n=1, 25, for(k=1, n, print1(binomial(n,k)*k^(n-k), ", "))) \\ G. C. Greubel, Jan 05 2017

# uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
bell_matrix(lambda n: n+1, 10) # Peter Luschny, Jan 18 2016

A059299 Triangle of idempotent numbers (version 3), T(n, k) = binomial(n, k) * (n - k)^k.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 6, 3, 0, 1, 12, 24, 4, 0, 1, 20, 90, 80, 5, 0, 1, 30, 240, 540, 240, 6, 0, 1, 42, 525, 2240, 2835, 672, 7, 0, 1, 56, 1008, 7000, 17920, 13608, 1792, 8, 0, 1, 72, 1764, 18144, 78750, 129024, 61236, 4608, 9, 0, 1, 90, 2880, 41160

Offset: 0

N. J. A. Sloane, Jan 25 2001

			Triangle begins:
1,
1,  0,
1,  2,   0,
1,  6,   3,    0,
1, 12,  24,    4,    0,
1, 20,  90,   80,    5,   0,
1, 30, 240,  540,  240,   6, 0,
1, 42, 525, 2240, 2835, 672, 7, 0,
...

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

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

There are 4 versions: A059297-A059300.
Diagonals give A001788, A036216, A040075, A050982, A002378, 3*A002417, etc.
Row sums are A000248.

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

T := (n, k) -> binomial(n, k) * (n - k)^k:
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;

t[n_, k_] := Binomial[n, k]*(n - k)^k; Prepend[Flatten@Table[t[n, k], {n, 10}, {k, 0, n}], 1] (* Arkadiusz Wesolowski, Mar 23 2013 *)

concat([1], for(n=0, 25, for(k=0, n, print1(binomial(n,k)*(n-k)^k, ", ")))) \\ G. C. Greubel, Jan 05 2017

Name corrected by Peter Luschny, Nov 12 2023

A154372 Triangle T(n,k) = (k+1)^(n-k)*binomial(n,k).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 12, 9, 1, 1, 32, 54, 16, 1, 1, 80, 270, 160, 25, 1, 1, 192, 1215, 1280, 375, 36, 1, 1, 448, 5103, 8960, 4375, 756, 49, 1, 1, 1024, 20412, 57344, 43750, 12096, 1372, 64, 1

Offset: 0

Paul Curtz, Jan 08 2009

From A152650/A152656,coefficients of other exponential polynomials(*). a(n) is triangle A152818 where terms of each column is divided by the beginning one. See A000004, A001787(n+1), A006043=2*A027472, A006044=6*A038846.
(*) Not factorial as written in A006044. See A000110, Bell-Touchard. Second diagonal is 1,4,9,16,25, denominators of Lyman's spectrum of hydrogen, A000290(n+1) which has homogeneous indices for denominators series of Rydberg-Ritz spectrum of hydrogen.
The matrix inverse starts
      1;
     -1,      1;
      3,     -4,     1;
    -16,     24,    -9,     1;
    125,   -200,    90,   -16,   1;
  -1296,   2160, -1080,   240, -25,   1;
  16807, -28812, 15435, -3920, 525, -36, 1;
.. compare with A122525 (row reversed). - R. J. Mathar, Mar 22 2013
From Peter Bala, Jan 14 2015: (Start)
Exponential Riordan array [exp(z), z*exp(z)]. This triangle is the particular case a = 0, b = 1, c = 1 of the triangle of generalized Stirling numbers of the second kind S(a,b,c) defined in the Bala link. Cf. A059297.
This is the triangle of connection constants when expressing the monomials x^n as a linear combination of the basis polynomials (x - 1)*(x - k - 1)^(k-1), k = 0,1,2,.... For example, from row 3 we have x^3 = 1 + 12*(x - 1) + 9*(x - 1)*(x - 3) + (x - 1)*(x - 4)^2.
Let M be the infinite lower unit triangular array with (n,k)-th entry (k*(n - k + 1) + 1)/(k + 1)*binomial(n,k). M is the row reverse of A145033. For k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0  M/ having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. The infinite product M(0)*M(1)*M(2)*..., which is clearly well-defined, is equal to the present triangle. See the Example section. (End)
T(n,k) is also the number of idempotent partial transformations of {1,2,...,n} having exactly k fixed points. - Geoffrey Critzer, Nov 25 2021

			With the array M(k) as defined in the Comments section, the infinite product M(0)*M(1)*M(2)*... begins
/1      \ /1        \ /1        \      /1        \
|1 1     ||0 1       ||0 1      |      |1  1      |
|1 3 1   ||0 1 1     ||0 0 1    |... = |1  4  1   |
|1 6 5 1 ||0 1 3 1   ||0 0 1 1  |      |1 12  9  1|
|...     ||0 1 6 5 1 ||0 0 1 3 1|      |...       |
|...     ||...       ||...      |      |          |
- _Peter Bala_, Jan 13 2015

G. C. Greubel, Table of n, a(n) for n = 0..1274
Peter Bala, A 3 parameter family of generalized Stirling numbers
Emanuele Munarini, Combinatorial identities involving the central coefficients of a Sheffer matrix, Applicable Analysis and Discrete Mathematics (2019) Vol. 13, 495-517.

Cf. A080108, A152818, A059297, A145033.

/* As triangle */ [[(k+1)^(n-k)*Binomial(n,k) : k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 15 2016

T[n_, k_] := (k + 1)^(n - k)*Binomial[n, k]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Sep 15 2016 *)

T(n,k) = (k+1)^(n-k)*binomial(n,k). k!*T(n,k) gives the entries for A152818 read as a triangular array.
E.g.f.: exp(x*(1+t*exp(x))) = 1 + (1+t)*x + (1+4*t+t^2)*x^2/2! + (1+12*t+9*t^2+t*3)*x^3/3! + .... O.g.f.: Sum_{k>=1} (t*x)^(k-1)/(1-k*x)^k = 1 + (1+t)*x + (1+4*t+t^2)*x^2 + .... Row sums are A080108. - Peter Bala, Oct 09 2011
From Peter Bala, Jan 14 2015: (Start)
Recurrence equation: T(n+1,k+1) = T(n,k+1) + Sum_{j = 0..n-k} (j + 1)*binomial(n,j)*T(n-j,k) with T(n,0) = 1 for all n.
Equals the matrix product A007318 * A059297. (End)

A216973 Exponential Riordan array [x*exp(x),x].

Original entry on oeis.org

0, 1, 0, 2, 2, 0, 3, 6, 3, 0, 4, 12, 12, 4, 0, 5, 20, 30, 20, 5, 0, 6, 30, 60, 60, 30, 6, 0, 7, 42, 105, 140, 105, 42, 7, 0, 8, 56, 168, 280, 280, 168, 56, 8, 0, 9, 72, 252, 504, 630, 504, 252, 72, 9, 0, 10, 90, 360, 840, 1260, 1260, 840, 360, 90, 10, 0

Offset: 0

Peter Bala, Sep 21 2012

This is the triangle of denominators from Leibniz's harmonic triangle, A003506, augmented with a main diagonal of 0's.

Note, the usual definition of the exponential Riordan array [f(x), x*g(x)] associated with a pair of power series f(x) and g(x) requires f(0) to be nonzero. Here we don't make this assumption. - Peter Bala, Feb 13 2017

			Triangle begins
.n\k.|..0.....1.....2.....3.....4.....5.....6
= = = = = = = = = = = = = = = = = = = = = = =
..0..|..0
..1..|..1.....0
..2..|..2.....2.....0
..3..|..3.....6.....3.....0
..4..|..4....12....12.....4.....0
..5..|..5....20....30....20.....5.....0
..6..|..6....30....60....60....30.....6.....0
...

A003506, A007318, A059297, A116071, A132440, A215652.

A216973_row := proc(n) x*exp(x)*exp(x*t): series(%,x,n+1): n!*coeff(%,x,n):
seq(coeff(%,t,k), k=0..n) end:
for n from 0 to 10 do A216973_row(n) od; # Peter Luschny, Feb 03 2017

(* The function RiordanArray is defined in A256893. *)
RiordanArray[# Exp[#]&, Identity, 11, True] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

T(n,k) = (n-k)*binomial(n,k) for 0 <= k <= n.
E.g.f.: x*exp(x)*exp(x*t) = 1 + x + (2 + 2*t)*x^2/2! + (3 + 6*t + 3*t^2)*x^3/3! + ....
The exponential Riordan array [x*exp(x),x] factors as [x,x]*[exp(x),x] = A132440*A007318.
This array is the infinitesimal generator for A116071; that is, Exp(A216973) = A116071, where Exp denotes the matrix exponential. A signed version of the array is the infinitesimal generator for A215652.
The first column of the array Exp(t*A216973) is the sequence of idempotent polynomials, the row polynomials of A059297.

A291203 Number F(n,h,t) of forests of t labeled rooted trees with n vertices such that h is the maximum of 0 and the tree heights; triangle of triangles F(n,h,t), n>=0, h=0..n, t=0..n-h, read by layers, then by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 1, 0, 3, 6, 0, 6, 0, 0, 0, 0, 0, 1, 0, 4, 24, 12, 0, 36, 24, 0, 24, 0, 0, 0, 0, 0, 0, 1, 0, 5, 80, 90, 20, 0, 200, 300, 60, 0, 300, 120, 0, 120, 0, 0, 0, 0, 0, 0, 0, 1, 0, 6, 240, 540, 240, 30, 0, 1170, 3000, 1260, 120, 0, 3360, 2520, 360, 0, 2520, 720, 0, 720, 0

Offset: 0

Alois P. Heinz, Aug 20 2017

Positive elements in column t=1 give A034855.
Elements in rows h=0 give A023531.
Elements in rows h=1 give A059297.
Positive row sums per layer give A235595.
Positive column sums per layer give A061356.

			n h\t: 0   1   2  3  4 5 : A235595 : A061356          : A000272
-----+-------------------+---------+------------------+--------
0 0  : 1                 :         :                  : 1
-----+-------------------+---------+------------------+--------
1 0  : 0   1             :      1  :   .              :
1 1  : 0                 :         :   1              : 1
-----+-------------------+---------+------------------+--------
2 0  : 0   0   1         :      1  :   .   .          :
2 1  : 0   2             :      2  :   .              :
2 2  : 0                 :         :   2   1          : 3
-----+-------------------+---------+------------------+--------
3 0  : 0   0   0  1      :      1  :   .   .   .      :
3 1  : 0   3   6         :      9  :   .   .          :
3 2  : 0   6             :      6  :   .              :
3 3  : 0                 :         :   9   6   1      : 16
-----+-------------------+---------+------------------+--------
4 0  : 0   0   0  0  1   :      1  :   .   .   .  .   :
4 1  : 0   4  24 12      :     40  :   .   .   .      :
4 2  : 0  36  24         :     60  :   .   .          :
4 3  : 0  24             :     24  :   .              :
4 4  : 0                 :         :  64  48  12  1   : 125
-----+-------------------+---------+------------------+--------
5 0  : 0   0   0  0  0 1 :      1  :   .   .   .  . . :
5 1  : 0   5  80 90 20   :    195  :   .   .   .  .   :
5 2  : 0 200 300 60      :    560  :   .   .   .      :
5 3  : 0 300 120         :    420  :   .   .          :
5 4  : 0 120             :    120  :   .              :
5 5  : 0                 :         : 625 500 150 20 1 : 1296
-----+-------------------+---------+------------------+--------

Alois P. Heinz, Layers n = 0..48, flattened

Cf. A000007, A000012, A000142, A000272, A000551, A001477, A001788, A001854, A002378, A023531, A034855, A038154, A059297, A061356, A089946, A126804, A234953, A235595, A235596, A243014, A291204, A291336, A291529.

b:= proc(n, t, h) option remember; expand(`if`(n=0 or h=0, x^(t*n), add(
       binomial(n-1, j-1)*j*x^t*b(j-1, 0, h-1)*b(n-j, t, h), j=1..n)))
    end:
g:= (n, h)-> b(n, 1, h)-`if`(h=0, 0, b(n, 1, h-1)):
F:= (n, h, t)-> coeff(g(n, h), x, t):
seq(seq(seq(F(n, h, t), t=0..n-h), h=0..n), n=0..8);

b[n_, t_, h_] := b[n, t, h] = Expand[If[n == 0 || h == 0, x^(t*n), Sum[
     Binomial[n-1, j-1]*j*x^t*b[j-1, 0, h-1]*b[n-j, t, h], {j, 1, n}]]];
g[n_, h_] := b[n, 1, h] - If[h == 0, 0, b[n, 1, h - 1]];
F[n_, h_, t_] := Coefficient[g[n, h], x, t];
Table[Table[Table[F[n, h, t], {t, 0, n - h}], {h, 0, n}], {n, 0, 8}] // Flatten (* Jean-François Alcover, Mar 17 2022, after Alois P. Heinz *)

Sum_{i=0..n} F(n,i,n-i) = A243014(n) = 1 + A038154(n).
Sum_{d=0..n} Sum_{i=0..d} F(n,i,d-i) = A000272(n+1).
Sum_{h=0..n} Sum_{t=0..n-h} t * F(n,h,t) = A089946(n-1) for n>0.
Sum_{h=0..n} Sum_{t=0..n-h} (h+1) * F(n,h,t) = A234953(n+1) for n>0.
Sum_{h=0..n} Sum_{t=0..n-h} (h+1)*(n+1) * F(n,h,t) = A001854(n+1) for n>0.
Sum_{t=0..n-1} F(n,1,t) = A235596(n+1).
F(2n,n,n) = A126804(n) for n>0.
F(n,0,n) = 1 = A000012(n).
F(n,1,1) = n = A001477(n) for n>1.
F(n,n-1,1) = n! = A000142(n) for n>0.
F(n,1,n-1) = A002378(n-1) for n>0.
F(n,2,1) = A000551(n).
F(n,3,1) = A000552(n).
F(n,4,1) = A000553(n).
F(n,1,2) = A001788(n-1) for n>2.
F(n,0,0) = A000007(n).

A256895 Triangle read by rows, T(n,k) = Sum_{j=0..n-k+1} j!C(n-1,j-1)T(n-j,k-1) if k != 0 else 1, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 11, 7, 1, 1, 49, 47, 13, 1, 1, 261, 341, 139, 21, 1, 1, 1631, 2731, 1471, 329, 31, 1, 1, 11743, 24173, 16213, 4789, 671, 43, 1, 1, 95901, 235463, 189373, 69441, 12881, 1231, 57, 1, 1, 876809, 2509621, 2357503, 1032245, 237961, 30169, 2087, 73, 1

Offset: 0

Peter Luschny, Apr 28 2015

Can be understood as a convolution matrix or as a sequence-to-triangle transformation similar to the partial Bell polynomials defined as: S -> T(n, k, S) = Sum_{j=0..n-k+1} C(n-1,j-1)*S(j)*T(n-j,k-1,S) if k != 0 else S(0)^n. Here S(n) = n!. The case S(n) = n gives the triangle of idempotent numbers A059297 and the case S(n) = 1 for all n leads to A256894.

			Triangle starts:
1;
1, 1;
1, 3, 1;
1, 11, 7, 1;
1, 49, 47, 13, 1;
1, 261, 341, 139, 21, 1;

Cf. A001339, A002061, A059297, A256894.

# Implemented as a sequence transformation acting on f: n -> n!.
F := proc(n, k, f) option remember; `if`(k=0, f(0)^n,
add(binomial(n-1, j-1)*f(j)*F(n-j, k-1, f), j=0..n-k+1)) end:
for n from 0 to 7 do seq(F(n, k, j->j!), k=0..n) od;

T(n+1,1) = A001339(n) for n>=0.

T(n,n-1) = A002061(n) for n>=1.

A359759 Table read by rows. T(n, k) = (-1)^(n - k) * Sum_{j=k..n} binomial(n, j) * A354794(j, k) * j^(n - j).

Original entry on oeis.org

1, 0, 1, 0, -3, 1, 0, 13, -9, 1, 0, -103, 79, -18, 1, 0, 1241, -905, 265, -30, 1, 0, -19691, 13771, -4290, 665, -45, 1, 0, 384805, -262885, 82621, -14630, 1400, -63, 1, 0, -8918351, 6007247, -1888362, 353381, -40390, 2618, -84, 1

Offset: 0

Peter Luschny, Jan 27 2023

Inspired by a formula of Mélika Tebni in A048993.

			Triangle T(n, k) starts:
[0] 1;
[1] 0,         1;
[2] 0,        -3,          1;
[3] 0,        13,         -9,        1;
[4] 0,      -103,         79,      -18,        1;
[5] 0,      1241,       -905,      265,      -30,       1;
[6] 0,    -19691,      13771,    -4290,      665,     -45,      1;
[7] 0,    384805,    -262885,    82621,   -14630,    1400,    -63,    1;
[8] 0,  -8918351,    6007247, -1888362,   353381,  -40390,   2618,  -84,    1;
[9] 0, 238966705, -159432369, 50110705, -9627702, 1206471, -96138, 4494, -108, 1;

Cf. A059297, A354794, A357247, A048993.

T := (n, k) -> (-1)^(n - k)*add(binomial(n, j) * A354794(j, k) * j^(n - j), j = k..n): for n from 0 to 9 do seq(T(n, k), k = 0..n) od;

E.g.f. of column k: (exp(LambertW(x*exp(-x))) - 1)^k / k!. (Note that (exp(-LambertW(-x*exp(-x))) - 1)^k / k! is the e.g.f. of column k of Stirling2.) - Mélika Tebni, Jan 27 2023

A367271 a(n) = binomial(2n, n) n^n.

Original entry on oeis.org

1, 2, 24, 540, 17920, 787500, 43110144, 2826399576, 215922769920, 18836384175180, 1847560000000000, 201267982422458952, 24110526523754151936, 3150082833623386551800, 445775933018180704665600, 67925014299030761718750000, 11087976122055526032070410240

Offset: 0

Peter Luschny, Nov 11 2023

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..322

Cf. A059297, A293270.

Maple
```
seq(binomial(2*n, n) * n^n, n = 0..16);
```

A367271[n_]:=If[n==0,1,Binomial[2n,n]n^n];
Array[A367271,20,0] (* Paolo Xausa, Nov 29 2023 *)

a(n) = A059297(2*n, n).

a(n) ~ 2^(2*n)*n^n/sqrt(n*Pi). - Stefano Spezia, Nov 29 2023

A367273 a(n) = Sum_{k=0..n} binomial(n, k)^2 * (k - n)^k.

Original entry on oeis.org

1, 1, -3, -8, 81, 26, -3815, 17494, 178241, -2817746, 3552201, 315952418, -3635118575, -11060115936, 782886068497, -7772807719574, -66097429593855, 2841563213504406, -26634464325602135, -375731325639156710, 14734035378180288401, -142992751647059748944

Offset: 0

Peter Luschny, Nov 11 2023

sign

Cf. A059297.

a := n -> add(binomial(n, k)^2 * (k - n)^k, k = 0..n):
seq(a(n), n = 0..22);

A367273[n_]:=If[n==0,1,Sum[Binomial[n,k]^2(k-n)^k,{k,0,n}]];
Array[A367273,30,0] (* Paolo Xausa, Nov 29 2023 *)

a(n) = Sum_{k=0..n} (-1)^k * binomial(n, k) * A059297(n, n - k).

A367272 a(n) = Sum_{k=0..n} binomial(n, k)^2 * k^(n - k).

Original entry on oeis.org

1, 1, 5, 28, 209, 1826, 18217, 203106, 2487361, 33077566, 473318201, 7234847126, 117435618577, 2014339775800, 36360190887217, 688237505878726, 13618646813974785, 280960214041690038, 6028928694559721305, 134277542969681115870, 3098232871805383942801

Offset: 0

Peter Luschny, Nov 11 2023

nonn

Cf. A059297.

a := n -> add(binomial(n, k)^2*k^(n - k), k = 0 .. n):
seq(a(n), n = 0..22);

Join[{1}, Table[Sum[Binomial[n,k]^2 * k^(n-k), {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Nov 12 2023 *)

a(n) = Sum_{k=0..n} binomial(n, k) * A059297(n, k).

log(a(n)) ~ n*(log(n) - log(log(n)) - 1 + (3*log(log(n)) + 2)/log(n) - 1/log(n)^2). - Vaclav Kotesovec, Nov 12 2023

cp's OEIS Frontend

A059298 Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 2.

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A256895 Triangle read by rows, T(n,k) = Sum_{j=0..n-k+1} j!*C(n-1,j-1)*T(n-j,k-1) if k != 0 else 1, n>=0, 0<=k<=n.

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