A001863 -id:A001863 - cp's OEIS frontend

A001865 Number of connected functions on n labeled nodes.

1, 3, 17, 142, 1569, 21576, 355081, 6805296, 148869153, 3660215680, 99920609601, 2998836525312, 98139640241473, 3478081490967552, 132705415800984825, 5423640496274200576, 236389784118231290049, 10944997108429625524224, 536484538620663729658993

Offset: 1

N. J. A. Sloane

If one randomly selects a ball from an urn containing n different balls, with replacement, until exactly one ball has been selected twice, the probability that that ball was also the first ball selected once is a(n)/n^n. See also A000435. - Matthew Vandermast, Jun 15 2004

a(n) equals the permanent of the (n-1) X (n-1) matrix with n+1's along the main diagonal and 1's everywhere else. - John M. Campbell, Apr 20 2012

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, p. 112.
Ulrike Sattler, Decidable classes of formal power series with nice closure properties, Diplomarbeit im Fach Informatik, Univ. Erlangen - Nuernberg, Jul 27 1994
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).

Alois P. Heinz, Table of n, a(n) for n = 1..380 (first 50 terms from T. D. Noe)
H. Bergeron, E. M. F. Curado, J. P. Gazeau and L. M. C. S. Rodrigues, A note about combinatorial sequences and Incomplete Gamma function, arXiv preprint arXiv: 1309.6910 [math.CO], 2013.
Christian Brouder, William J. Keith, and Ângela Mestre, Closed forms for a multigraph enumeration, arXiv preprint arXiv:1301.0874 [math.CO], 2013.
Giulio Cerbai and Anders Claesson, Counting fixed-point-free Cayley permutations, arXiv:2507.09304 [math.CO], 2025. See pp. 8, 19.
Camille Combe, A geometric and combinatorial exploration of Hochschild lattices, arXiv:2007.00048 [math.CO], 2020. See p. 22.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 37
Leo Katz, Probability of indecomposability of a random mapping function, Ann. Math. Statist. 26, (1955), 512-517.
John Riordan, Letter to N. J. A. Sloane, Aug. 1970
Frank Schmidt and Rodica Simion, Card shuffling and a transformation on S_n, Aequationes Math. 44 (1992), no. 1, 11-34.
Bernd Sturmfels and Ngoc Mai Tran, Combinatorial Types of Tropical Eigenvectors, arXiv:1105.5504 [math.CO], 2011-2012.

a(n) = A000435(n) + n^(n-1). See also A063169.

Column k=1 of A060281.

spec := [B, {A=Prod(Z,Set(A)), B=Cycle(A)}, labeled]; [seq(combstruct[count](spec,size=n), n=0..20)];
seq(simplify(GAMMA(n,n)*exp(n)),n=1..20); # Vladeta Jovovic, Jul 21 2005

t=Sum[n^(n-1)x^n/n!,{n,1,20}];
Range[0,20]! CoefficientList[Series[Log[1/(1-t)]+1,{x,0,20}],x] (* Geoffrey Critzer, Mar 12 2011 *)
f[n_] := Sum[n! n^(n - k - 1)/(n - k)!, {k, n}]; Array[f, 18] (* Robert G. Wilson v *)
a[n_] := Exp[n]*Gamma[n, n]; Table[a[n] // FunctionExpand, {n, 1, 18}] (* Jean-François Alcover, May 13 2013, after Vladeta Jovovic *)

a(n)=if(n<0,0,n!*sum(k=1,n,n^(n-k-1)/(n-k)!))

a(n)=(1/n)*sum(k=1,n,binomial(n,k)*(n-k)^(n-k)*k^k) \\ Paul D. Hanna, Jul 04 2013

N=20; x='x+O('x^N); Vec(serlaplace(log(sum(k=0, N, (k*x)^k/k!)))) \\ Seiichi Manyama, May 27 2019

from math import comb
def A001865(n): return ((sum(comb(n,k)*(n-k)**(n-k)*k**k for k in range(1,(n+1>>1)))<<1) + (0 if n&1 else comb(n,m:=n>>1)*m**n))//n + n**(n-1) # Chai Wah Wu, Apr 25-26 2023

a(n) = Sum_{k=1..n} n!*n^(n-k-1) / (n-k)!.
E.g.f.: -log(1+LambertW(-x)). - Vladeta Jovovic, Apr 11 2001
E.g.f. satisfies 0=2y'^4+2y''^2-y'''y'-y''y'^2. - Michael Somos, Aug 23 2003
Integral representation in terms of the incomplete Gamma function: a(n) = exp(n+1)*Gamma(n+1,n+1) = exp(n+1)*Integral_{x=n+1..oo} x^n exp(-x) dx.
Asymptotics: sqrt(Pi*n/2)*n^(n-1). - N-E. Fahssi, Jan 25 2008, corrected by Vaclav Kotesovec, Nov 27 2012
a(n) = exp(1)*Integral_{x=1..oo} (n+x)^n*exp(-x) dx. - Gerald McGarvey, Apr 16 2008
a(n) = (1/n) * Sum_{k=1..n} C(n,k) * (n-k)^(n-k) * k^k. - Paul D. Hanna, Jul 04 2013
From Peter Bala, Jun 29 2016: (Start)
It appears that a(n) = (n-1)!*( e^n - Sum_{k >= 0} n^(n + k)/(n + k)! ) = (n-1)!*( e^n - Sum_{k >= 0} k^2*n^(n + k - 1)/(n + k)! ).
Note that (n-1)!*( e^n - Sum_{k >= 0} k^3*n^(n + k - 1)/(n + k)! ) also appears to be an integer sequence beginning [1, 5, 37, 370, 4681, 71736, 1292005, ...]. (End)
a(n) = Sum_{k=1..n} (n!/(n-k)!) * k^2 * n^(n-k-2). - Brian P Hawkins, Feb 07 2024

More terms from James Sellers, May 23 2000

A000435 Normalized total height of all nodes in all rooted trees with n labeled nodes.

Original entry on oeis.org

0, 1, 8, 78, 944, 13800, 237432, 4708144, 105822432, 2660215680, 73983185000, 2255828154624, 74841555118992, 2684366717713408, 103512489775594200, 4270718991667353600, 187728592242564421568, 8759085548690928992256, 432357188322752488126152, 22510748754252398927872000

Offset: 1

N. J. A. Sloane

This is the sequence that started it all: the first sequence in the database!
The height h(V) of a node V in a rooted tree is its distance from the root. a(n) = Sum_{all nodes V in all n^(n-1) rooted trees on n nodes} h(V)/n.
In the trees which have [0, n-1] = (0, 1, ..., n-1) as their ordered set of nodes, the number of nodes at distance i from node 0 is f(n,i) = (n-1)...(n-i)(i+1)n^(j-1), 0 <= i < n-1, i+j = n-1 (and f(n,n-1) = (n-1)!): (n-1)...(n-i) counts the words coding the paths of length i from any node to 0, n^(j-1) counts the Pruefer codes of the rest, words build by iterated deletion of the greater node of degree 1 ... except the last one, (i+1), necessary pointing at the path. If g(n,i) = (n-1)...(n-i)n^j, i+j = n-1, f(n,i) = g(n,i) - g(n,i+1), g(n,i) = Sum_{k>=i} f(n,k), the sequence is Sum_{i=1..n-1} g(n,i). - Claude Lenormand (claude.lenormand(AT)free.fr), Jan 26 2001
If one randomly selects one ball from an urn containing n different balls, with replacement, until exactly one ball has been selected twice, the probability that this ball was also the second ball to be selected once is a(n)/n^n. See also A001865. - Matthew Vandermast, Jun 15 2004
a(n) is the number of connected endofunctions with no fixed points. - Geoffrey Critzer, Dec 13 2011
a(n) is the number of weakly connected simple digraphs on n labeled nodes where every node has out-degree 1. A digraph where all out-degrees are 1 can be called a functional digraph due to the correspondence with endofunctions. - Andrew Howroyd, Feb 06 2024

			For n = 3 there are 3^2 = 9 rooted labeled trees on 3 nodes, namely (with o denoting a node, O the root node):
   o
   |
   o     o   o
   |      \ /
   O       O
The first can be labeled in 6 ways and contains nodes at heights 1 and 2 above the root, so contributes 6*(1+2) = 18 to the total; the second can be labeled in 3 ways and contains 2 nodes at height 1 above the root, so contributes 3*2=6 to the total, giving 24 in all. Dividing by 3 we get a(3) = 24/3 = 8.
For n = 4 there are 4^3 = 64 rooted labeled trees on 4 nodes, namely (with o denoting a node, O the root node):
   o
   |
   o     o        o   o
   |     |         \ /
   o     o   o      o     o o o
   |      \ /       |      \|/
   O       O        O       O
  (1)     (2)      (3)     (4)
Tree (1) can be labeled in 24 ways and contains nodes at heights 1, 2, 3 above the root, so contributes 24*(1+2+3) = 144 to the total;
tree (2) can be labeled in 24 ways and contains nodes at heights 1, 1, 2 above the root, so contributes 24*(1+1+2) = 96 to the total;
tree (3) can be labeled in 12 ways and contains nodes at heights 1, 2, 2 above the root, so contributes 12*(1+2+2) = 60 to the total;
tree (4) can be labeled in 4 ways and contains nodes at heights 1, 1, 1 above the root, so contributes 4*(1+1+1) = 12 to the total;
giving 312 in all. Dividing by 4 we get a(4) = 312/4 = 78.

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).

Robert G. Wilson v, Table of n, a(n) for n = 1..1000 (first 100 terms from T. D. Noe)
Vijayakumar Ambat, Article in the Malayalam newspaper Ayala Manorama - Padhippura, 12 June 2015, that mentions the OEIS, and in particular this sequence.
V. I. Arnold, Topological classification of complex trigonometric polynomials and the combinatorics of graphs with the same number of edges and vertices, Functional Anal. Appl., 30 (1996), 1-17.
Shalosh B. Ekhad and Doron Zeilberger, Going Back to Neil Sloane's FIRST LOVE (OEIS Sequence A435): On the Total Heights in Rooted Labeled Trees, arXiv:1607.05776 [math.CO], 2016.
Shalosh B. Ekhad and Doron Zeilberger, Going Back to Neil Sloane's FIRST LOVE (OEIS Sequence A435): On the Total Heights in Rooted Labeled Trees, Version on DZ's home page with more links; Local copy, pdf file only, no active links
I. P. Goulden and D. M. Jackson, A proof of a conjecture for the number of ramified coverings of the sphere by the torus, arXiv:math/9902009 [math.AG], 1999.
I. P. Goulden, D. M. Jackson, and A. Vainshtein, The number of ramified coverings of the sphere by the torus and surfaces of higher genera, arXiv:math/9902125 [math.AG], 1999.
I. P. Goulden, D. M. Jackson, and A. Vainshtein, The number of ramified coverings of the sphere by the torus and surfaces of higher genera Ann. Comb. 4 (2000), no. 1, 27-46. (See Theorem 1.1.)
Brady Haran, The Number Collector (with Neil Sloane), Numberphile Podcast (2019)
Andrew Lohr and Doron Zeilberger, On the limiting distributions of the total height on families of trees, Integers (2018) 18, Article #A32.
T. Kyle Petersen, Exponential generating functions and Bell numbers, Inquiry-Based Enumerative Combinatorics (2019) Chapter 7, Undergraduate Texts in Mathematics, Springer, Cham, 98-99.
A. Rényi and G. Szekeres, On the height of trees, Journal of the Australian Mathematical Society 7.04 (1967): 497-507. See (4.7).
Marko Riedel et al., Connected endofunctions with no fixed points, Mathematics Stack Exchange, Dec 2014.
J. Riordan, Letter to N. J. A. Sloane, Aug. 1970
J. Riordan and N. J. A. Sloane, Enumeration of rooted trees by total height, J. Austral. Math. Soc., vol. 10 pp. 278-282, 1969.
N. J. A. Sloane, Page from 1964 notebook showing start of OEIS [includes A000027, A000217, A000292, A000332, A000389, A000579, A000110, A007318, A000058, A000215, A000289, A000324, A234953 (= A001854(n)/n), A000435, A000169, A000142, A000272, A000312, A000111]
N. J. A. Sloane, Cover of same notebook
N. J. A. Sloane, Lengths of Cycle Times in Random Neural Networks, Ph. D. Dissertation, Cornell University, February 1967; also Report No. 10, Cognitive Systems Research Program, Cornell University, 1967. This sequence appears on page 119.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, Illustration of a(3) and a(4)
Yukun Yao and Doron Zeilberger, An Experimental Mathematics Approach to the Area Statistic of Parking Functions, arXiv:1806.02680 [math.CO], 2018.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 3.
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Cf. A001863, A001864, A001854, A002862 (unlabeled version), A234953, A259334.

Column k=1 of A350452.

A000435 := n-> (n-1)!*add (n^k/k!, k=0..n-2);
seq(simplify((n-1)*GAMMA(n-1,n)*exp(n)),n=1..20); # Vladeta Jovovic, Jul 21 2005

f[n_] := (n - 1)! Sum [n^k/k!, {k, 0, n - 2}]; Array[f, 18] (* Robert G. Wilson v, Aug 10 2010 *)
nx = 18; Rest[ Range[0, nx]! CoefficientList[ Series[ LambertW[-x] - Log[1 + LambertW[-x]], {x, 0, nx}], x]] (* Robert G. Wilson v, Apr 13 2013 *)

x='x+O('x^30); concat(0, Vec(serlaplace(lambertw(-x)-log(1+lambertw(-x))))) \\ Altug Alkan, Sep 05 2018

A000435(n)=(n-1)*A001863(n) \\ M. F. Hasler, Dec 10 2018

from math import comb
def A000435(n): return ((sum(comb(n,k)*(n-k)**(n-k)*k**k for k in range(1,(n+1>>1)))<<1) + (0 if n&1 else comb(n,m:=n>>1)*m**n))//n # Chai Wah Wu, Apr 25-26 2023

a(n) = (n-1)! * Sum_{k=0..n-2} n^k/k!.
a(n) = A001864(n)/n.
E.g.f.: LambertW(-x) - log(1+LambertW(-x)). - Vladeta Jovovic, Apr 10 2001
a(n) = A001865(n) - n^(n-1).
a(n) = A001865(n) - A000169(n). - Geoffrey Critzer, Dec 13 2011
a(n) ~ sqrt(Pi/2)*n^(n-1/2). - Vaclav Kotesovec, Aug 07 2013
a(n)/A001854(n) ~ 1/2 [See Renyi-Szekeres, (4.7)]. Also a(n) = Sum_{k=0..n-1} k*A259334(n,k). - David desJardins, Jan 20 2017
a(n) = (n-1)*A001863(n). - M. F. Hasler, Dec 10 2018

Additional references from Valery A. Liskovets
Editorial changes by N. J. A. Sloane, Feb 03 2012
Edited by M. F. Hasler, Dec 10 2018

A001864 Total height of rooted trees with n labeled nodes.

Original entry on oeis.org

0, 2, 24, 312, 4720, 82800, 1662024, 37665152, 952401888, 26602156800, 813815035000, 27069937855488, 972940216546896, 37581134047987712, 1552687346633913000, 68331503866677657600, 3191386068123595166656, 157663539876436721860608

Offset: 1

N. J. A. Sloane

a(n) is the total number of nonrecurrent elements mapped into a recurrent element in all functions f:{1,2,...,n}->{1,2,...,n}. a(n) = Sum_{k=1..n-1} A216971(n,k)*k. - Geoffrey Critzer, Jan 01 2013

a(n) is the sum of the lengths of all cycles over all functions f:{1,2,...,n}->{1,2,...,n}. Fixed points are taken to have length zero. a(n) = Sum_{k=2..n} A066324(n,k)*(k-1). - Geoffrey Critzer, Aug 19 2013

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
Hien D. Nguyen and G. J. McLachlan, Progress on a Conjecture Regarding the Triangular Distribution, arXiv preprint arXiv:1607.04807 [stat.OT], 2016.
J. Riordan, Letter to N. J. A. Sloane, Aug. 1970
J. Riordan and N. J. A. Sloane, Enumeration of rooted trees by total height, J. Austral. Math. Soc., vol. 10 pp. 278-282, 1969.
N. J. A. Sloane, Illustration of terms a(3) and a(4) in A000435
D. Zvonkine, Home Page
D. Zvonkine, An algebra of power series arising in the intersection theory of moduli spaces of curves and in the enumeration of ramified coverings of the sphere, arXiv:0403092v2 [math.AG], 2004.
D. Zvonkine, Enumeration of ramified coverings of the sphere and 2-dimensional gravity, arXiv:math/0506248 [math.AG], 2005.
D. Zvonkine, Counting ramified coverings and intersection theory on Hurwitz spaces II (local structure of Hurwitz spaces and combinatorial results), Moscow Mathematical Journal, vol. 7 (2007), no. 1, 135-162.
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Cf. A000435, A001863.

A001864 := proc(n) local k; add(n!*n^k/k!, k=0..n-2); end;

Table[Sum[Binomial[n,k](n-k)^(n-k) k^k,{k,1,n-1}],{n,20}] (* Harvey P. Dale, Oct 10 2011 *)
a[n_] := n*(n-1)*Exp[n]*Gamma[n-1, n] // Round; Table[a[n], {n, 1, 18}]  (* Jean-François Alcover, Jun 24 2013 *)

a(n)=sum(k=1,n-1,binomial(n,k)*(n-k)^(n-k)*k^k)

from math import comb
def A001864(n): return (sum(comb(n,k)*(n-k)**(n-k)*k**k for k in range(1,(n+1>>1)))<<1) + (0 if n&1 else comb(n,m:=n>>1)*m**n) # Chai Wah Wu, Apr 25-26 2023

a(n) = n*A000435(n).
E.g.f: (LambertW(-x)/(1+LambertW(-x)))^2. - Vladeta Jovovic, Apr 10 2001
a(n) = Sum_{k=1..n-1} binomial(n, k)*(n-k)^(n-k)*k^k. - Benoit Cloitre, Mar 22 2003
a(n) ~ sqrt(Pi/2)*n^(n+1/2). - Vaclav Kotesovec, Aug 07 2013
a(n) = n! * Sum_{k=0..n-2} n^k/k!. - Jianing Song, Aug 08 2022

A001866 Number of connected graphs with n nodes and n edges.

Original entry on oeis.org

0, 0, 1, 24, 936, 56640, 4968000, 598328640, 94916183040, 19200422062080, 4826695329792000, 1476585999504000000, 540272647694971699200, 233019960215154829516800, 117009251702203840384204800, 67680314823703303654732800000, 44677678066673631080900198400000

Offset: 0

N. J. A. Sloane

nonn

Or number of n X n (0,1) matrices with two 1's in each row the permanent of which equals to 2. Note that, if (0,1) matrix with two 1's in each row has positive permanent, then it is equal to a power of 2. - Vladimir Shevelev, Mar 25 2010

V. S. Shevelev, On the permanent of the stochastic (0,1)-matrices with equal row sums, Izvestia Vuzov of the North-Caucasus region, Nature sciences 1 (1997), 21-38 (in Russian). - Vladimir Shevelev, Mar 25 2010
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 = 0..100
T. L. Austin, R. E. Fagen, W. F. Penney, and John Riordan, The number of components in random linear graphs, Ann. Math. Statist 30 1959 747-754.
J. Riordan, Letter to N. J. A. Sloane, Aug. 1970

Cf. A174586.

Join[{0}, Table[(n!^2*n^(n - 1)/2)*Sum[n^(-k)/(n - k)!, {k, 2, n}], {n, 20}]] (* T. D. Noe, Aug 10 2012 *)

Explicit formula: a(n) = (n!^2*n^(n-1)/2)*Sum_{k=2..n} n^(-k)/(n-k)!; Recursion: a(2)=1, for n>=3, a(n) = n!*((n-1)!/2+Sum_{k=2..n-1} (-1)^(n+k+1)*k^(n-k)*binomial(n,k)*a(k)/k!). - Vladimir Shevelev, Mar 25 2010

a(n) ~ Pi * n^(2*n) / (2*exp(n)). - Vaclav Kotesovec, Nov 30 2017

A320064 The number of F_2 graphs on { 1, 2, ..., n } with a unique cycle of weight 1, which corresponds to the number of reflectable bases of the root system of type D_n.

Original entry on oeis.org

0, 1, 16, 312, 7552, 220800, 7597824, 301321216, 13545271296, 681015214080, 37879390720000, 2309968030334976, 153275504883695616, 10995166075754119168, 847974316241667686400, 69971459959477921382400, 6151490510604350965940224, 574035430519008722436489216, 56669921387839814123670994944

Offset: 1

Masaya Tomie, Oct 04 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..350
Federico Ardila, Matthias Beck, and Jodi McWhirter, The Arithmetic of Coxeter Permutahedra, arXiv:2004.02952 [math.CO], 2020.
S. Azam, M. B. Soltani, M. Tomie and Y. Yoshii, A graph theoretical classification for reflectable bases, PRIMS, Vol 55 no 4, (2019), 689-736.
Theo Douvropoulos, Joel Brewster Lewis, and Alejandro H. Morales, Hurwitz numbers for reflection groups III: Uniform formulas, arXiv:2308.04751 [math.CO], 2023, see p. 11.

Cf. A000435, A001863, A038057.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (&+[(&+[j^(j-1)*(2*x)^j/Factorial(j): j in [1..m+2]])^k/(4*k): k in [2..m+1]]) )); [0] cat [Factorial(n+1)*b[n]: n in [1..m-2]]; // G. C. Greubel, Dec 10 2018

nmax = 20; Rest[CoefficientList[Series[Sum[1/(4*m)*(Sum[k^(k-1)*(2*x)^k/k!, {k, 1, nmax}])^m, {m, 2, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!] (* Vaclav Kotesovec, Oct 23 2018 *)

seq(n)={Vec(serlaplace(sum(m=2, n, (sum(k=1, n, k^(k-1)*(2*x)^k/k!) + O(x^n))^m/(4*m))), -n)} \\ Andrew Howroyd, Nov 07 2018

apply( A320064(n)=A001863(n)*(n-1)<<(n-2), [1..20]) \\ Defines the function A320064. The additional apply(...) provides a check and illustration. - M. F. Hasler, Dec 09 2018

from math import comb
def A320064(n): return 0 if n<2 else ((sum(comb(n,k)*(n-k)**(n-k)*k**k for k in range(1,(n+1>>1)))<<1) + (0 if n&1 else comb(n,m:=n>>1)*m**n))//n<Chai Wah Wu, Apr 25-26 2023

E.g.f.: Sum_{m>=2} (1/(4*m)) (Sum_{k>=1} k^(k-1)*(2*x)^k/k!)^m.
a(n) = (n-1)*2^(n-2)*A001863(n). - M. F. Hasler, Dec 09 2018
a(n) = 2^(n-2)*A000435(n). - Chai Wah Wu, Apr 25 2023

A321233 a(n) is the number of reflectable bases of the root system of type D_n.

Original entry on oeis.org

0, 4, 128, 4992, 241664, 14131200, 972521472, 77138231296, 6935178903552, 697359579217920, 77576992194560000, 9461629052252061696, 1255632936007234486272, 180144800985155488448512, 27786422394606966747955200, 4585649599904345055716966400, 806288164205933489807717040128

Offset: 1

Masaya Tomie, Nov 01 2018

nonn

The root systems of type D_n are only defined for n >= 4. See chapter 3 of the Humphreys reference. Sequence extended to n=1 using formula/recurrence.

J. E. Humphreys, Introduction to Lie algebras and representation theory, 2nd ed, Springer-Verlag, New York, 1972.

S. Azam, M. B. Soltani, M. Tomie and Y. Yoshii, A graph theoretical classification for reflectable bases, PRIMS, Vol 55 no 4, (2019), 689-736.

Cf. A000435, A320064.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (&+[ (&+[ j^(j-1)*(4*x)^j/Factorial(j) :j in [1..m+3]])^k/(4*k) :k in [2..m+2]]) )); [0] cat [Factorial(n+1)*b[n]: n in [1..m-2]]; // G. C. Greubel, Dec 09 2018

Rest[With[{m = 25}, CoefficientList[Series[Sum[Sum[j^(j - 1)*(4*x)^j/j!, {j, 1, m + 1}]^k/(4*k), {k, 2, m}], {x, 0, m}], x]*Range[0, m]!]] (* G. C. Greubel, Dec 09 2018 *)

a(n)={n!*polcoef(sum(m=2, n, (sum(k=1, n, k^(k-1)*(4*x)^k/k!) + O(x^(n-m+2)))^m/(4*m)), n)} \\ Andrew Howroyd, Nov 01 2018

A321233(n)=A001863(n)*(n-1)*4^(n-1) \\ M. F. Hasler, Dec 09 2018

from math import comb
def A321233(n): return 0 if n<2 else ((sum(comb(n,k)*(n-k)**(n-k)*k**k for k in range(1,(n+1>>1)))<<1) + (0 if n&1 else comb(n,m:=n>>1)*m**n))//n<<(n-1<<1) # Chai Wah Wu, Apr 26 2023

E.g.f.: Sum_{m>=2} (1/(4*m)) (Sum_{k>=1} k^(k-1)*(4*x)^k/k!)^m.
a(n) = 2^n*A320064(n).
a(n) = (n-1)*4^(n-1)*A001863(n). - M. F. Hasler, Dec 09 2018

A134558 Array read by antidiagonals, a(n,k) = gamma(n+1,k)*e^k, where gamma(n,k) is the upper incomplete gamma function and e is the exponential constant 2.71828...

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 6, 5, 3, 1, 24, 16, 10, 4, 1, 120, 65, 38, 17, 5, 1, 720, 326, 168, 78, 26, 6, 1, 5040, 1957, 872, 393, 142, 37, 7, 1, 40320, 13700, 5296, 2208, 824, 236, 50, 8, 1, 362880, 109601, 37200, 13977, 5144, 1569, 366, 65, 9, 1, 3628800, 986410, 297856

Offset: 0

Ross La Haye, Jan 22 2008

			Square array begins:
    1,    1,    1,     1,     1,     1,      1, ...
    1,    2,    3,     4,     5,     6,      7, ...
    2,    5,   10,    17,    26,    37,     50, ...
    6,   16,   38,    78,   142,   236,    366, ...
   24,   65,  168,   393,   824,  1569,   2760, ...
  120,  326,  872,  2208,  5144, 10970,  21576, ...
  720, 1957, 5296, 13977, 34960, 81445, 176112, ...

Eric Weisstein's World of Mathematics, Incomplete Gamma Function.
Wikipedia, Incomplete gamma function.

Cf. a(n, 0) = A000142(n); a(n, 1) = A000522(n); a(n, 2) = A010842(n); a(n, 3) = A053486(n); a(n, 4) = A053487(n); a(n, 5) = A080954(n); a(n, 6) = A108869(n); a(1, k) = A000027(k+1); a(2, k) = A002522(k+1); a(n, n) = A063170(n); a(n, n+1) = A001865(n+1); a(n, n+2) = A001863(n+2).
Another version: A089258.
A transposed version: A080955.
Cf. A001113.

T[n_,k_] := Gamma[n+1, k]*E^k; Table[T[n-k, k], {n, 0, 10}, {k, 0, n}] //Flatten (* Amiram Eldar, Jun 27 2020 *)

a(n,k) = gamma(n+1,k)*e^k = Sum_{m=0..n} m!*binomial(n,m)*k^(n-m).
a(n,k) = n*a(n-1,k) + k^n for n,k > 0.
E.g.f. (by columns) is e^(kx)/(1-x).
a(n,k) = the binomial transform by columns of a(n,k-1).
Conjecture: a(n,k) is the permanent of the n X n matrix with k+1 on the main diagonal and 1 elsewhere.

More terms from Amiram Eldar, Jun 27 2020

cp's OEIS Frontend

A177453 Partial sums of A001863.

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A001865 Number of connected functions on n labeled nodes.

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A000435 Normalized total height of all nodes in all rooted trees with n labeled nodes.

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A001864 Total height of rooted trees with n labeled nodes.

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A001866 Number of connected graphs with n nodes and n edges.

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A320064 The number of F_2 graphs on { 1, 2, ..., n } with a unique cycle of weight 1, which corresponds to the number of reflectable bases of the root system of type D_n.

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