A137452 -id:A137452 - cp's OEIS frontend

A177885 a(n) = (1-n)^(n-1).

1, 1, -1, 4, -27, 256, -3125, 46656, -823543, 16777216, -387420489, 10000000000, -285311670611, 8916100448256, -302875106592253, 11112006825558016, -437893890380859375, 18446744073709551616, -827240261886336764177

Offset: 0

Vladimir Kruchinin, Dec 28 2010

A signed version of A000312.

LeClair gives an approximation z(n) for the location of the n-th nontrivial zero of the Riemann zeta function on the critical line, which can be expressed in terms of the exponential generating function of this sequence A(x) = x/LambertW(x) as follows: z(n) = 1/2 + 2*Pi*exp(1)*A((n - 11/8)/exp(1))*i. For example, working to 1 decimal place, z(1) = 1/2 + 14.5*i (the first nontrivial zero is at 1/2 + 14.1*i), z(10) = 1/2 + 50.2*i (the tenth nontrivial zero is at 1/2 + 49.8*i) and z(100) = 1/2 + 236*i (the hundredth nontrivial zero is at 1/2 + 236.5*i). [Peter Bala, Jun 12 2013]

			From _Paul D. Hanna_, Aug 24 2016: (Start)
E.g.f.: A(x) = 1 + x - x^2/2! + 4*x^3/3! - 27*x^4/4! + 256*x^5/5! - 3125*x^6/6! + 46656*x^7/7! - 823543*x^8/8! +...+ (1-n)^(n-1)*x^n/n! +...
Related series.
Series_Reversion(A(x) - 1) = x + x^2/2 - x^3/6 + x^4/12 - x^5/20 + x^6/30 - x^7/42 + x^8/56 - x^9/72 + x^10/90 +...+ (-x)^n/(n*(n-1)) +... (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..140
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See pp. 32, 34.
Vladimir Kruchinin and Dmitry V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.
André LeClair, An electrostatic depiction of the validity of the Riemann Hypothesis and a formula for the N-th zero at large N, arXiv:1305.2613 [math-ph], 2013.

Cf. A000312, A137452 (row sums).

[(1-n)^(n-1): n in [0..30]]; // Vincenzo Librandi, May 15 2011

Join[{1,1}, Table[(1-n)^(n-1), {n, 2, 20}]] (* Harvey P. Dale, Aug 10 2012 *)
nn = 18; Range[0, nn]! CoefficientList[ Series[ Exp[ ProductLog[ x]], {x, 0, nn}], x] (* Robert G. Wilson v, Aug 23 2012 *)

a(n)=(1-n)^(n-1) \\ Charles R Greathouse IV, May 15 2013

{a(n) = my(A = 1 + serreverse( x + sum(m=2,n+2, (-x)^m/(m*(m-1)) +x^2*O(x^n)))); n!*polcoeff(A,n)}
for(n=0,30,print1(a(n),", ")) \\ Paul D. Hanna, Aug 24 2016

E.g.f. satisfies A(x) = exp(x/A(x)).
E.g.f. A(x) = x/LambertW(x) = exp(LambertW(x)) = 1 + x - x^2/2! + 4*x^3/3! - 27*x^4/4! + .... - Peter Bala, Jun 12 2013
E.g.f.: 1 + Series_Reversion( (1+x)*log(1+x) ). - Paul D. Hanna, Aug 24 2016
E.g.f.: 1 + Series_Reversion( x + Sum_{n>=2} (-x)^n/(n*(n-1)) ). - Paul D. Hanna, Aug 24 2016
a(n) ~ (-1)^(n+1) * exp(-1) * n^(n-1). - Vaclav Kotesovec, Sep 22 2016
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n-1, k-1)*n^(n-k), for n >= 1 and a(0) = 1, that is, Sum_{k=0..n}*A137452(n, k), for n >= 0. - Wolfdieter Lang, Apr 11 2023

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

Original entry on oeis.org

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

A049444 Generalized Stirling number triangle of first kind.

Original entry on oeis.org

1, -2, 1, 6, -5, 1, -24, 26, -9, 1, 120, -154, 71, -14, 1, -720, 1044, -580, 155, -20, 1, 5040, -8028, 5104, -1665, 295, -27, 1, -40320, 69264, -48860, 18424, -4025, 511, -35, 1, 362880, -663696, 509004, -214676, 54649, -8624, 826, -44, 1, -3628800, 6999840, -5753736

Offset: 0

Wolfdieter Lang

T(n, k) = ^2P_n^k in the notation of the given reference with T(0, 0) := 1. The monic row polynomials s(n,x) := Sum_{m=0..n} T(n, k)*x^k which are s(n, x) = Product_{j=0..n-1} (x-(2+j)), n >= 1 and s(0, x)=1 satisfy s(n, x+y) = Sum_{k=0..n} binomial(n, k)*s(k,x)*S1(n-k, y), with the Stirling1 polynomials S1(n, x) = Sum_{m=1..n} (A008275(n, m)*x^m) and S1(0, x)=1.
In the umbral calculus (see the S. Roman reference given in A048854) the s(n, x) polynomials are called Sheffer polynomials for (exp(2*t), exp(t)-1). This translates to the usual exponential Riordan (Sheffer) notation (1/(1+x)^2, log(1+x)).
See A143491 for the unsigned version of this array and A143494 for the inverse. - Peter Bala, Aug 25 2008
Corresponding to the generalized Stirling number triangle of second kind A137650. - Peter Luschny, Sep 18 2011
Unsigned, reversed rows (cf. A145324, A136124) are the dimensions of the cohomology of a complex manifold with a symmetric group (S_n) action. See p. 17 of the Hyde and Lagarias link. See also the Murri link for an interpretation as the Betti numbers of the moduli space M(0,n) of smooth Riemann surfaces. - Tom Copeland, Dec 09 2016
The row polynomials s(n, x) = (-1)^n*risingfactorial(2 - x, n) are related to the column sequences of the unsigned Abel triangle A137452(n, k), for k >= 2. See the formula there. - Wolfdieter Lang, Nov 21 2022

			The Triangle  begins:
n\k       0       1        2       3       4      5      6    7   8 9 ...
0:        1
1:       -2       1
2:        6      -5        1
3:      -24      26       -9       1
4:      120    -154       71     -14       1
5      -720    1044     -580     155     -20      1
6:     5040   -8028     5104   -1665     295    -27      1
7:   -40320   69264   -48860   18424   -4025    511    -35    1
8:   362880 -663696   509004 -214676   54649  -8624    826  -44
9: -3628800 6999840 -5753736 2655764 -761166 140889 -16884 1266 -54 1
...  [reformatted by _Wolfdieter Lang_, Nov 21 2022]

Y. Manin, Frobenius Manifolds, Quantum Cohomology and Moduli Spaces, American Math. Soc. Colloquium Publications Vol. 47, 1999. [From Tom Copeland, Jun 29 2008]
S. Roman, The Umbral Calculus, Academic Press, 1984 (also Dover Publications, 2005).

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
E. Getzler, Operads and moduli spaces of genus 0 Riemann surfaces, arXiv:alg-geom/9411004, 1994, (see p. 23, g(x,t)). [From _Tom Copeland_, Dec 11 2011]
T. Hyde and J. Lagarias Polynomial splitting measures and cohomology of the pure braid group, arXiv preprint arXiv:1604.05359 [math.RT], 2016.
Y. Manin, Generating functions in algebraic geometry and sums over trees, arXiv:alg-geom/9407005, 1994, (Eqn. 0.7 and 1.7). [From _Tom Copeland_, Dec 10 2011]
D. S. Mitrinovic and M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 (1962), 1-77.
R. Murri, Fatgraph Algorithms and the Homology of the Kontsevich Complex, arXiv:1202.1820 [math.AG], 2012, (see Table 1, p. 3). [From _Tom Copeland_, Sep 18 2012]

Unsigned column sequences are A000142(n+1), A001705-A001709. Row sums (signed triangle): n!*(-1)^n, row sums (unsigned triangle): A001710(n-2). Cf. A008275 (Stirling1 triangle).
Cf. A000035, A084938, A094645, A094646.
Cf. A143491, A143494.
Cf. A136124, A137452, A137650.

a049444 n k = a049444_tabl !! n !! k
a049444_row n = a049444_tabl !! n
a049444_tabl = map fst $ iterate (\(row, i) ->
   (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 2)
-- Reinhard Zumkeller, Mar 11 2014

A049444_row := proc(n) local k,i;
add(add(Stirling1(n, n-i), i=0..k)*x^(n-k-1),k=0..n-1);
seq(coeff(%,x,k),k=1..n-1) end:
seq(print(A049444_row(n)),n=1..7); # Peter Luschny, Sep 18 2011
A049444:= (n, k)-> add((-1)^(n-j)*(n-j+1)!*binomial(n, j)*Stirling1(j, k), j=0..n):
seq(print(seq(A049444(n, k), k=0..n)), n=0..11);  # Mélika Tebni, May 02 2022

t[n_, i_] = Sum[(-1)^k*Binomial[n, k]*(k+1)!*StirlingS1[n-k, i], {k, 0, n-i}]; Flatten[Table[t[n, i], {n, 0, 9}, {i, 0, n}]] [[1 ;; 48]]
(* Jean-François Alcover, Apr 29 2011, after Milan Janjic *)

T(n, k) = T(n-1, k-1) - (n+1)*T(n-1, k), n >= k >= 0; T(n, k) = 0, n < k; T(n, -1) = 0, T(0, 0) = 1.
E.g.f. for k-th column of signed triangle: ((log(1+x))^k)/(k!*(1+x)^2).
Triangle (signed) = [-2, -1, -3, -2, -4, -3, -5, -4, -6, -5, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...]; triangle (unsigned) = [2, 1, 3, 2, 4, 3, 5, 4, 6, 5, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...], where DELTA is Deléham's operator defined in A084938 (unsigned version in A143491).
E.g.f.: (1 + x)^(y-2). - Vladeta Jovovic, May 17 2004 [For row polynomials s(n, y)]
With P(n, t) = Sum_{j=0..n-2} T(n-2,j) * t^j and P(1, t) = -1 and P(0, t) = 1, then G(x, t) = -1 + exp[P(.,t)*x] = [(1+x)^t - 1 - t^2 * x] / [t(t-1)], whose compositional inverse in x about 0 is given in A074060. G(x, 0) = -log(1+x) and G(x, 1) = (1+x) log(1+x) - 2x. G(x, q^2) occurs in formulas on pages 194-196 of the Manin reference. - Tom Copeland, Feb 17 2008
If we define f(n,i,a) = Sum_{k=0..n-i} binomial(n,k)*Stirling1(n-k,i)*Product_{j=0..k-1} (-a-j), then T(n,i) = f(n,i,2), for n=1,2,...; i=0..n. - Milan Janjic, Dec 21 2008
T(n, k) = Sum_{j=0..n} (-1)^(n-j)*(n-j+1)!*binomial(n, j)*Stirling1(j, k). - Mélika Tebni, May 02 2022
From Wolfdieter Lang, Nov 24 2022: (Start)
Recurrence for row polynomials {s(n, x)}_{n>=0}: s(0, x) = 1, s(n, x) = (x - 2)*exp(-(d/dx)) s(n-1, x), for n >= 1. This is adapted from the general Sheffer result given by S. Roman, Corollary 3.7.2., p. 50.
Recurrence for column sequence {T(n, k)}{n>=k}: T(n, n) = 1, T(n, k) = (n!/(n-k))*Sum{j=k..n-1} (1/j!)*(a(n-1-j) + k*beta(n-1-j))*T(n-1, k), for k >= 0, where alpha = repeat(-2, 2) and beta(n) = [x^n] (d/dx)log(log(x)/x) = (-1)^(n+1)*A002208(n+1)/A002209(n+1), for n >= 0. This is the adapted Boas-Buck recurrence, also given in Rainville, Theorem 50., p. 141, For the references and a comment see A046521. (End)

Second formula corrected by Philippe Deléham, Nov 09 2008

A061356 Triangle read by rows: T(n, k) is the number of labeled trees on n nodes with maximal node degree k (0 < k < n).

Original entry on oeis.org

1, 2, 1, 9, 6, 1, 64, 48, 12, 1, 625, 500, 150, 20, 1, 7776, 6480, 2160, 360, 30, 1, 117649, 100842, 36015, 6860, 735, 42, 1, 2097152, 1835008, 688128, 143360, 17920, 1344, 56, 1, 43046721, 38263752, 14880348, 3306744, 459270, 40824, 2268, 72, 1

Offset: 2

Olivier Gérard, Jun 07 2001

Essentially the coefficients of the Abel polynomials (A137452). - Peter Luschny, Jun 12 2022
This is a formula from Comtet, Theorem F, vol. I, p. 81 (French edition) used in proving Theorem D.
If we let N = n+1, binomial(N-2, k-1)*(N-1)^(N-k-1) = binomial(n-1, k-1)*n^(n-k), so this sequence with offset 1,1 also gives the number of rooted forests of k trees over [n]. - Washington Bomfim, Jan 09 2008
Let S(n,k) be the signed triangle, S(n,k) = (-1)^(n-k)T(n,k), which starts 1, -2, 1, 9, -6, 1, ..., then the inverse of S is the triangle of idempotent numbers A059298. - Peter Luschny, Mar 13 2009
With offset 1 also number of labeled multigraphs of k components, n nodes, and no cycles except one loop in each component. See link below to have a picture showing the bijection between rooted forests and multigraphs of this kind. (Note that there are no labels in the picture, but the bijection remains true if we label the nodes.) - Washington Bomfim, Sep 04 2010
With offset 1, T(n,k) is the number of forests of rooted trees on n nodes with exactly k (rooted) trees. - Geoffrey Critzer, Feb 10 2012
Also the Bell transform of the sequence (n+1)^n (A000169(n+1)) without column 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 21 2016
Abel polynomials A(n,x) = x*(x+n)^(n-1) satisfy d/dx A(n,x) = n*A(n-1,x+1). - Michael Somos, May 10 2024
Also, T(n,k) is the number of parking functions with k ties. - Kyle Celano, Aug 18 2025

			Triangle begins
    1;
    2,     1;
    9,     6,     1;
   64,    48,    12,    1;
  625,   500,   150,   20,    1;
 7776,  6480,  2160,  360,   30,    1;
 ...
From _Peter Bala_, Sep 21 2012: (Start)
O.g.f.'s for the diagonals begin:
1/(1-x) = 1 + x + x^2 + x^3 + ...
2*x/(1-x)^3 = 2 + 6*x + 12*x^3 + ... A002378(n+1)
(9+3*x)/(1-x)^5 = 9 + 48*x + 150*x^2 + ... 3*A004320(n+1)
The numerator polynomials are the row polynomials of A155163.
(End)

L. Comtet, Analyse Combinatoire, P.U.F., Paris 1970. Volume 1, p 81.
L. Comtet, Advanced Combinatorics, Reidel, 1974.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
A. Avron and N. Dershowitz, Cayley's Formula, A Page From The Book, Amer. Math. Monthly, Vol. 123, No. 7, Aug.-Sep. 2016, 699-700 (2).
Peter Bala, Diagonals of triangles with generating function exp(t*F(x))
W. Bomfim, Bijection between rooted forests and multigraphs without cycles except one loop in each connected component. [From _Washington Bomfim_, Sep 04 2010]
J. W. Moon, Another proof of Cayley's formula for counting trees, Amer. Math. Monthly, Vol. 70, No. 8, Oct. 1963, 846-847.
Jim Pitman, Coalescent Random Forests, Technical Report No. 457, Department of Statistics, University of California.
Jim Pitman, Coalescent Random Forests, Journal of Combinatorial Theory, Series A, Volume 85, Issue 2, February 1999, Pages 165-193.
Paul R. F. Schumacher, Descents in parking functions, J. Integer Seq., Vol. 21 (2018), Article 18.2.3.
Eric Weisstein's World of Mathematics, Abel Polynomial.
Wikipedia, Lambert W function
J. Zeng, A Ramanujan sequence that refines the Cayley formula for trees, Ramanujan J., 3(1999) 1, 45-54.

Variant of A137452.
Columns are A000169, A053506, A053507, A053508, A053509.
First diagonal is A002378.
Row sums give A000272.
Cf. A028421, A059297, A139526 (row reverse), A155163, A202017.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0,...) as column 0 to the triangle.
BellMatrix(n -> (n+1)^n, 12); # Peter Luschny, Jan 21 2016

nn = 7; t = Sum[n^(n - 1)  x^n/n!, {n, 1, nn}]; f[list_] := Select[list, # > 0 &]; Map[f, Drop[Range[0, nn]! CoefficientList[Series[Exp[y t], {x, 0, nn}], {x, y}], 1]] // Flatten  (* Geoffrey Critzer, Feb 10 2012 *)
T[n_, m_] := T[n, m] = Binomial[n, m]*Sum[m^k*T[n-m, k], {k, 1, n-m}]; T[n_, n_] = 1; Table[T[n, m], {n, 1, 9}, {m, 1, n}] // Flatten (* Jean-François Alcover, Mar 31 2015, after Vladimir Kruchinin *)
Table[Binomial[n - 2, k - 1]*(n - 1)^(n - k - 1), {n, 2, 12}, {k, 1, n - 1}] // Flatten (* G. C. Greubel, Nov 12 2017 *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];
rows = 10;
M = BellMatrix[(# + 1)^#&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

create_list(binomial(n,k)*(n+1)^(n-k),n,0,20,k,0,n); /* Emanuele Munarini, Apr 01 2014 */

for(n=2,11, for(k=1,n-1, print1(binomial(n-2, k-1)*(n-1)^(n-k-1), ", "))) \\ G. C. Greubel, Nov 12 2017

# uses[bell_matrix from A264428]
# Adds (1,0,0,0,...) as column 0 to the triangle.
bell_matrix(lambda n: (n+1)^n, 12) # Peter Luschny, Jan 21 2016

T(n, k) = binomial(n-2, k-1)*(n-1)^(n-k-1).
E.g.f.: (-LambertW(-y)/y)^(x+1)/(1+LambertW(-y)). - Vladeta Jovovic
From Peter Bala, Sep 21 2012: (Start)
Let T(x) = Sum_{n >= 0} n^(n-1)*x^n/n! denote the tree function of A000169. E.g.f.: F(x,t) := exp(t*T(x)) - 1 = -1 + {T(x)/x}^t = t*x + t*(2 + t)*x^2/2! + t*(9 + 6*t + t^2)*x^3/3! + ....
The compositional inverse with respect to x of (1/t)*F(x,t) is the e.g.f. for a signed version of the row reverse of A028421.
The row generating polynomials are the Abel polynomials A(n,x) = x*(x+n)^(n-1) for n >= 1.
Define B(n,x) = x^n/(1+n*x)^(n+1) = (-1)^n*A(-n,-1/x) for n >= 1. The k-th column entries are the coefficients in the formal series expansion of x^k in terms of B(n,x). For example, Col. 1: x = B(1,x) + 2*B(2,x) + 9*B(3,x) + 64*B(4,x) + ..., Col. 2: x^2 = B(2,x) + 6*B(3,x) + 48*B(4,x) + 500*B(5,x) + ... Compare with A059297.
n-th row sum = A000272(n+1).
Row reverse triangle is A139526.
The o.g.f.'s for the diagonals of the triangle are the rational functions R(n,x)/(1-x)^(2*n+1), where R(n,x) are the row polynomials of A155163. See below for examples.
(End)
T(n,m) = C(n,m)*Sum_{k=1..n-m} m^k*T(n-m,k), T(n,n) = 1. - Vladimir Kruchinin, Mar 31 2015

A354794 Triangle read by rows. The Bell transform of the sequence {m^m | m >= 0}.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 4, 3, 1, 0, 27, 19, 6, 1, 0, 256, 175, 55, 10, 1, 0, 3125, 2101, 660, 125, 15, 1, 0, 46656, 31031, 9751, 1890, 245, 21, 1, 0, 823543, 543607, 170898, 33621, 4550, 434, 28, 1, 0, 16777216, 11012415, 3463615, 688506, 95781, 9702, 714, 36, 1

Offset: 0

Peter Luschny, Jun 09 2022

For the definition of the Bell transform see A264428. The Bell transform of {(-m)^m | m >= 0} is A039621. The numbers A039621(n, k) are known as the Lehmer-Comtet numbers of 2nd kind. We think it is more natural to use Bell_{n, k}({m^m}) as the basis for the definition (and let the triangle start at (0, 0)).

			Triangle T(n, k) begins:
[0] 1;
[1] 0,        1;
[2] 0,        1,        1;
[3] 0,        4,        3,       1;
[4] 0,       27,       19,       6,      1;
[5] 0,      256,      175,      55,     10,     1;
[6] 0,     3125,     2101,     660,    125,    15,    1;
[7] 0,    46656,    31031,    9751,   1890,   245,   21,   1;
[8] 0,   823543,   543607,  170898,  33621,  4550,  434,  28,  1;
[9] 0, 16777216, 11012415, 3463615, 688506, 95781, 9702, 714, 36, 1;

Louis Comtet, Advanced Combinatorics. Reidel, Dordrecht, 1974, p. 139-140.

D. H. Lehmer, Numbers Associated with Stirling Numbers and x^x, Rocky Mountain J. Math., 15(2) 1985, pp. 461-475.
Peter Luschny, The Bell transform
Wikipedia, Bell polynomials.

Cf. A264428, A039621 (signed variant), A195979 (row sums), A000312 (column 1), A045531 (column 2), A281596 (column 3), A281595 (column 4), A000217 (diagonal 1), A215862 (diagonal 2), A354795 (matrix inverse), A137452 (Abel).

T := (n, k) -> if n = k then 1 else
add((-1)^j*(n-j-1)^(n-1)/(j!*(k-1-j)!), j = 0.. k-1) fi:
seq(seq(T(n, k), k = 0..n), n = 0..9);
# Alternatively, using the function BellMatrix from A264428:
BellMatrix(n -> n^n, 9);
# Or by recursion:
R := proc(n, k, m) option remember;
   if k < 0 or n < 0 then 0 elif k = 0 then 1 else
   m*R(n, k-1, m) + R(n-1, k, m+1) fi end:
A039621 := (n, k) -> ifelse(n = 0, 1, R(k-1, n-k, n-k)):

Unprotect[Power]; Power[0, 0] = 1; pow[n_] := n^n;
R = Range[0, 9]; T[n_, k_] := BellY[n, k, pow[R]];
Table[T[n, k], {n, R}, {k, 0, n}] // Flatten

from functools import cache
@cache
def t(n, k, m):
    if k < 0 or n < 0: return 0
    if k == 0: return n ** k
    return m * t(n, k - 1, m) + t(n - 1, k, m + 1)
def A354794(n, k): return t(k - 1, n - k, n - k) if n != k else 1
for n in range(9): print([A354794(n, k) for k in range(n + 1)])

T(n, k) = Bell_{n, k}(A000312), where Bell_{n, k} is the partial Bell polynomial evaluated over the powers m^m (with 0^0 = 1). See the Mathematica program.
T(n, k) = Sum_{j=0..k-1} (-1)^j*(n-j-1)^(n - 1)/(j! * (k-1-j)!) for 0 <= k < n and T(n, n) = 1.
T(n, k) = r(k-1, n-k, n-k) for n,k >= 1 and T(0, 0) = 1, where r(n, k, m) = m*r(n, k-1, m) + r(n-1, k, m+1) and r(n, 0, m) = 1. (see Vladimir Kruchinin's formula in A039621).
Sum_{k=1..n} binomial(k + x - 1, k-1)*(k-1)!*T(n, k) = (n + x)^(n - 1) for n >= 1.
Sum_{k=1..n} (-1)^(k+j)*Stirling1(k, j)*T(n, k) = n^(n-j)*binomial(n-1, j-1) for n >= 1, which are, up to sign, the coefficients of the Abel polynomials (A137452).
From Werner Schulte, Jun 14 2022 and Jun 19 2022: (Start)
E.g.f. of column k >= 0: (Sum_{i>0} (i-1)^(i-1) * t^i / i!)^k / k!.
Conjecture: T(n, k) = Sum_{i=0..n-k} A048994(n-k, i) * A048993(n+i-1, n-1) for 0 < k <= n and T(n, 0) = 0^n for n >= 0; proved by Mike Earnest, see link at A354797. (End)

A232006 Triangular array read by rows: T(n,k) is the number of simple labeled graphs on vertex set {1,2,...,n} with exactly k components (all of which are trees) such that the labels {1,2,...,k} are all in distinct components (trees), n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 16, 8, 3, 1, 0, 125, 50, 15, 4, 1, 0, 1296, 432, 108, 24, 5, 1, 0, 16807, 4802, 1029, 196, 35, 6, 1, 0, 262144, 65536, 12288, 2048, 320, 48, 7, 1, 0, 4782969, 1062882, 177147, 26244, 3645, 486, 63, 8, 1, 0, 100000000, 20000000, 3000000, 400000, 50000, 6000, 700, 80, 9, 1

Offset: 0

Geoffrey Critzer, Nov 16 2013

Row sums = (n^n-n)/(n-1)^2 = A058128(n).

Column k without leading zeros is the k-th exponential (also called binomial) convolution of the sequence {A000272(n+1)} = {A232006(n+1, 1)}, for n >= 0, with e.g.f. LamberW(-x)/(-x), where LambertW is the principal branch of the Lambert W-function. This is also the row polynomial P(n, x) of the unsigned triangle A137452, evaluated at x = k. - Wolfdieter Lang, Apr 24 2023

			The triangle begins:
n\k  0         1        2       3      4     5    6   7  8 9 10 ...
0:   1
1:   0         1
2:   0         1        1
3:   0         3        2       1
4:   0        16        8       3      1
5:   0       125       50      15      4     1
6:   0      1296      432     108     24     5    1
7:   0     16807     4802    1029    196    35    6   1
8:   0    262144    65536   12288   2048   320   48   7  1
9:   0   4782969  1062882  177147  26244  3645  486  63  8 1
10:  0 100000000 20000000 3000000 400000 50000 6000 700 80 9  1
... Reformatted by _Wolfdieter Lang_, Apr 24 2023

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Proposition 5.3.2.

G. C. Greubel, Rows n=0..75 of triangle, flattened
Chad Casarotto, Graph Theory and Cayley's Formula, 2006
Alan D. Sokal, A remark on the enumeration of rooted labeled trees, arXiv:1910.14519 [math.CO], 2019.
Marc van Leeuwen, I am stuck with a combinatoric problem... Math Stackexchange, Answer May 14 2017.
Eric Weisstein's World of Mathematics, Lambert W-function
Wikipedia, Lambert W function

Cf. A058128, |A137452|.

Columns give A000007, A000272, A007334, A362354, A362355, A362356, ...

Prepend[Table[Table[k n^(n-k-1),{k,0,n}],{n,1,8}],{1}]//Grid

{T(n, k) = if( k<0 || k>n, 0, n^(n-k-1))}; /* Michael Somos, May 15 2017 */

T(n, k) = k*n^(n-k-1).
T(n, k) = Sum_{i=0..n-k} T(n-1, k-1+i)*C(n-k,i), T(0, 0) = 1, T(n, 0) = 0 when n >= 1.
From Wolfdieter Lang, Apr 24 2023: (Start)
E.g.f. for {T(n+k, k)}_{n>=0} is (LambertW(-x)/(-x))^k, for k >= 0.
T(n, k) = Sum_{m=0..n-k} |A137452(n-k, m)|*k^m, for n >= 0 and k = 0..n. That is, T(n, n) = 1, for n >= 0, and T(n, k) = Sum_{m=1..n-k} binomial(n-k-1, m-1)*(n-k)^(n-k-m)*k^m, for k = 0..n-1 and n >= k+1. (End)

A202477 The number of ways to build all endofunctions on each block of every set partition of {1,2,...,n}.

Original entry on oeis.org

1, 1, 5, 40, 437, 6036, 100657, 1965160, 43937385, 1106488720, 30982333661, 954607270464, 32090625710365, 1168646120904640, 45826588690845705, 1924996299465966976, 86231288506425806033, 4103067277186778016000, 206655307175847710248885

Offset: 0

Geoffrey Critzer, Dec 19 2011

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..385

Cf. A000262 (the same for permutations), A137452.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(i^(i*j)*b(n-i*j, i-1)*
       multinomial(n, n-i*j, i$j)/j!, j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 29 2016

nn = 20; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}] ;
Range[0, nn]! CoefficientList[Series[Exp[t/(1 - t)], {x, 0, nn}], x]

E.g.f.: exp(T(x)/(1-T(x))) where T(x) is the e.g.f. for A000169.
a(n) ~ n^(n-1/3) * exp(3/2*n^(1/3) - 2/3) / sqrt(3). - Vaclav Kotesovec, Sep 24 2013
a(n) = Sum_{k=0..n} n^(n-k)*binomial(n-1,k-1)*A000262(k). - Fabian Pereyra, Jul 12 2024
The above formula can be written with the Abel polynomials: a(n) = Sum_{k=0..n} (-1)^(n - k) * A137452(n, k) * A000262(k). - Peter Luschny, Jul 13 2024

A362354 a(n) = 3*(n+3)^(n-1).

Original entry on oeis.org

1, 3, 15, 108, 1029, 12288, 177147, 3000000, 58461513, 1289945088, 31813498119, 867763964928, 25949267578125, 844424930131968, 29713734098717811, 1124440102746243072, 45543381089624394897, 1966080000000000000000, 90125827485245075684223, 4372496892684322588065792

Offset: 0

Wolfdieter Lang, Apr 24 2023

This gives the third exponential (also called binomial) convolution of {A000272(n+1)} = {A232006(n+1, 1)}, for n >= 0, with e.g.f. (LambertW(-x),(-x)) (LambertW is the principal branch of the Lambert W-function).

This is also the row polynomial P(n, x) of the unsigned triangle A137452, evaluated at x = 3.

Eric Weisstein's World of Mathematics, Lambert W-function
Wikipedia, Lambert W function

Column k=3 of A232006 (without leading zeros).

Cf. A137452.

a(n) = Sum_{k=0..n} |A137452(n, k)|*3^k = Sum_{k=0..n} binomial(n-1, k-1)*n^(n-k)*3^k, with the n = 0 term equal to 1 (not 0).
E.g.f.: (LambertW(-x)/(-x))^3.
From Seiichi Manyama, Jun 19 2024: (Start)
E.g.f. A(x) satisfies:
(1) A(x) = exp(3*x*A(x)^(1/3)).
(2) A(x) = 1/A(-x*A(x)^(2/3)). (End)

A225465 Triangular array read by rows: T(n, k) is the number of rooted forests on {1, 2, ..., n} in which one tree has been specially designated that contain exactly k trees; n >= 1, 1 <= k <= n.

Original entry on oeis.org

1, 2, 2, 9, 12, 3, 64, 96, 36, 4, 625, 1000, 450, 80, 5, 7776, 12960, 6480, 1440, 150, 6, 117649, 201684, 108045, 27440, 3675, 252, 7, 2097152, 3670016, 2064384, 573440, 89600, 8064, 392, 8, 43046721, 76527504, 44641044, 13226976, 2296350, 244944, 15876, 576, 9

Offset: 1

Geoffrey Critzer, May 08 2013

Row sums = 2n*(n+1)^(n-2) = A089946(offset).
The average number of trees in each forest approaches 5/2 as n gets large.
The rows give the coefficients of the derivatives of the Abel polynomials. - Peter Luschny, Feb 22 2025

			    T(2,1)=2                  T(2,2)=2
  ...1'...   ...2'...   ...1'..2...   ...1..2'...
  ...| ...   ...| ...   ...........   ...........
  ...2 ...   ...1 ...   ...........   ...........
The root node is on top.  The ' indicates the tree which has been specially designated.
Triangle starts:
  [1]        1;
  [2]        2,        2;
  [3]        9,       12,        3;
  [4]       64,       96,       36,        4;
  [5]      625,     1000,      450,       80,       5;
  [6]     7776,    12960,     6480,     1440,     150,      6;
  [7]   117649,   201684,   108045,    27440,    3675,    252,     7;
  [8]  2097152,  3670016,  2064384,   573440,   89600,   8064,   392,   8;
  [9] 43046721, 76527504, 44641044, 13226976, 2296350, 244944, 15876, 576, 9;

Cf. A061356, A089946 (row sums), A000169, A137452.

Table[Table[Binomial[n - 1, k - 1] n^(n - k) k, {k, 1, n}], {n, 1, 8}] // Grid

T(n, k) = binomial(n-1, k-1)*n^(n-k)*k = A061356(n, k)*k(offset).

E.g.f.: y*A(x)*exp(y*A(x)) where A(x) is e.g.f. for A000169.

A362355 a(n) = 4*(n+4)^(n-1).

Original entry on oeis.org

1, 4, 24, 196, 2048, 26244, 400000, 7086244, 143327232, 3262922884, 82644187136, 2306601562500, 70368744177664, 2330488948919044, 83291859462684672, 3196026743131536484, 131072000000000000000, 5722274760967941313284, 264999811677837732610048

Offset: 0

Wolfdieter Lang, Apr 24 2023

This gives the fourth exponential (also called binomial) convolution of {A000272(n+1)} = {A232006(n+1, 1)}, for n >= 0, with e.g.f. (LambertW(-x),(-x)) (LambertW is the principal branch of the Lambert W-function).

This is also the row polynomial P(n, x) of the unsigned triangle A137452, evaluated at x = 4.

Eric Weisstein's World of Mathematics, Lambert W-function
Wikipedia, Lambert W function

Column k = 4 of A232006 (without leading zeros).

Cf. A000272, A137452.

Table[4(n+4)^(n-1),{n,0,20}] (* Harvey P. Dale, Jun 05 2024 *)

a(n) = Sum_{k=0..n} |A137452(n, k)|*4^k = Sum_{k=0..n} binomial(n-1, k-1)*n^(n-k)*4^k, with the n = 0 term equal to 1 (not 0).
E.g.f.: (LambertW(-x)/(-x))^4.
From Seiichi Manyama, Jun 19 2024: (Start)
E.g.f. A(x) satisfies:
(1) A(x) = exp(4*x*A(x)^(1/4)).
(2) A(x) = 1/A(-x*A(x)^(1/2)). (End)

cp's OEIS Frontend

A177885 a(n) = (1-n)^(n-1).

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

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A049444 Generalized Stirling number triangle of first kind.

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A061356 Triangle read by rows: T(n, k) is the number of labeled trees on n nodes with maximal node degree k (0 < k < n).

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A354794 Triangle read by rows. The Bell transform of the sequence {m^m | m >= 0}.

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A232006 Triangular array read by rows: T(n,k) is the number of simple labeled graphs on vertex set {1,2,...,n} with exactly k components (all of which are trees) such that the labels {1,2,...,k} are all in distinct components (trees), n >= 0, 0 <= k <= n.

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