A059297 -id:A059297 - cp's OEIS frontend

A002378 Oblong (or promic, pronic, or heteromecic) numbers: a(n) = n*(n+1).

0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462, 506, 552, 600, 650, 702, 756, 812, 870, 930, 992, 1056, 1122, 1190, 1260, 1332, 1406, 1482, 1560, 1640, 1722, 1806, 1892, 1980, 2070, 2162, 2256, 2352, 2450, 2550

Offset: 0

N. J. A. Sloane

4*a(n) + 1 are the odd squares A016754(n).
The word "pronic" (used by Dickson) is incorrect. - Michael Somos
According to the 2nd edition of Webster, the correct word is "promic". - R. K. Guy
a(n) is the number of minimal vectors in the root lattice A_n (see Conway and Sloane, p. 109).
Let M_n denote the n X n matrix M_n(i, j) = (i + j); then the characteristic polynomial of M_n is x^(n-2) * (x^2 - a(n)*x - A002415(n)). - Benoit Cloitre, Nov 09 2002
The greatest LCM of all pairs (j, k) for j < k <= n for n > 1. - Robert G. Wilson v, Jun 19 2004
First differences are a(n+1) - a(n) = 2*n + 2 = 2, 4, 6, ... (while first differences of the squares are (n+1)^2 - n^2 = 2*n + 1 = 1, 3, 5, ...). - Alexandre Wajnberg, Dec 29 2005
25 appended to these numbers corresponds to squares of numbers ending in 5 (i.e., to squares of A017329). - Lekraj Beedassy, Mar 24 2006
A rapid (mental) multiplication/factorization technique -- a generalization of Lekraj Beedassy's comment: For all bases b >= 2 and positive integers n, c, d, k with c + d = b^k, we have (n*b^k + c)*(n*b^k + d) = a(n)*b^(2*k) + c*d. Thus the last 2*k base-b digits of the product are exactly those of c*d -- including leading 0(s) as necessary -- with the preceding base-b digit(s) the same as a(n)'s. Examples: In decimal, 113*117 = 13221 (as n = 11, b = 10 = 3 + 7, k = 1, 3*7 = 21, and a(11) = 132); in octal, 61*67 = 5207 (52 is a(6) in octal). In particular, for even b = 2*m (m > 0) and c = d = m, such a product is a square of this type. Decimal factoring: 5609 is immediately seen to be 71*79. Likewise, 120099 = 301*399 (k = 2 here) and 99990000001996 = 9999002*9999998 (k = 3). - Rick L. Shepherd, Jul 24 2021
Number of circular binary words of length n + 1 having exactly one occurrence of 01. Example: a(2) = 6 because we have 001, 010, 011, 100, 101 and 110. Column 1 of A119462. - Emeric Deutsch, May 21 2006
The sequence of iterated square roots sqrt(N + sqrt(N + ...)) has for N = 1, 2, ... the limit (1 + sqrt(1 + 4*N))/2. For N = a(n) this limit is n + 1, n = 1, 2, .... For all other numbers N, N >= 1, this limit is not a natural number. Examples: n = 1, a(1) = 2: sqrt(2 + sqrt(2 + ...)) = 1 + 1 = 2; n = 2, a(2) = 6: sqrt(6 + sqrt(6 + ...)) = 1 + 2 = 3. - Wolfdieter Lang, May 05 2006
Nonsquare integers m divisible by ceiling(sqrt(m)), except for m = 0. - Max Alekseyev, Nov 27 2006
The number of off-diagonal elements of an (n + 1) X (n + 1) matrix. - Artur Jasinski, Jan 11 2007
a(n) is equal to the number of functions f:{1, 2} -> {1, 2, ..., n + 1} such that for a fixed x in {1, 2} and a fixed y in {1, 2, ..., n + 1} we have f(x) <> y. - Aleksandar M. Janjic and Milan Janjic, Mar 13 2007
Numbers m >= 0 such that round(sqrt(m+1)) - round(sqrt(m)) = 1. - Hieronymus Fischer, Aug 06 2007
Numbers m >= 0 such that ceiling(2*sqrt(m+1)) - 1 = 1 + floor(2*sqrt(m)). - Hieronymus Fischer, Aug 06 2007
Numbers m >= 0 such that fract(sqrt(m+1)) > 1/2 and fract(sqrt(m)) < 1/2 where fract(x) is the fractional part (fract(x) = x - floor(x), x >= 0). - Hieronymus Fischer, Aug 06 2007
X values of solutions to the equation 4*X^3 + X^2 = Y^2. To find Y values: b(n) = n(n+1)(2n+1). - Mohamed Bouhamida, Nov 06 2007
Nonvanishing diagonal of A132792, the infinitesimal Lah matrix, so "generalized factorials" composed of a(n) are given by the elements of the Lah matrix, unsigned A111596, e.g., a(1)*a(2)*a(3) / 3! = -A111596(4,1) = 24. - Tom Copeland, Nov 20 2007
If Y is a 2-subset of an n-set X then, for n >= 2, a(n-2) is the number of 2-subsets and 3-subsets of X having exactly one element in common with Y. - Milan Janjic, Dec 28 2007
a(n) coincides with the vertex of a parabola of even width in the Redheffer matrix, directed toward zero. An integer p is prime if and only if for all integer k, the parabola y = kx - x^2 has no integer solution with 1 < x < k when y = p; a(n) corresponds to odd k. - Reikku Kulon, Nov 30 2008
The third differences of certain values of the hypergeometric function 3F2 lead to the squares of the oblong numbers i.e., 3F2([1, n + 1, n + 1], [n + 2, n + 2], z = 1) - 3*3F2([1, n + 2, n + 2], [n + 3, n + 3], z = 1) + 3*3F2([1, n + 3, n + 3], [n + 4, n + 4], z = 1) - 3F2([1, n + 4, n + 4], [n + 5, n + 5], z = 1) = (1/((n+2)*(n+3)))^2 for n = -1, 0, 1, 2, ... . See also A162990. - Johannes W. Meijer, Jul 21 2009
Generalized factorials, [a.(n!)] = a(n)*a(n-1)*...*a(0) = A010790(n), with a(0) = 1 are related to A001263. - Tom Copeland, Sep 21 2011
For n > 1, a(n) is the number of functions f:{1, 2} -> {1, ..., n + 2} where f(1) > 1 and f(2) > 2. Note that there are n + 1 possible values for f(1) and n possible values for f(2). For example, a(3) = 12 since there are 12 functions f from {1, 2} to {1, 2, 3, 4, 5} with f(1) > 1 and f(2) > 2. - Dennis P. Walsh, Dec 24 2011
a(n) gives the number of (n + 1) X (n + 1) symmetric (0, 1)-matrices containing two ones (see [Cameron]). - L. Edson Jeffery, Feb 18 2012
a(n) is the number of positions of a domino in a rectangled triangular board with both legs equal to n + 1. - César Eliud Lozada, Sep 26 2012
a(n) is the number of ordered pairs (x, y) in [n+2] X [n+2] with |x-y| > 1. - Dennis P. Walsh, Nov 27 2012
a(n) is the number of injective functions from {1, 2} into {1, 2, ..., n + 1}. - Dennis P. Walsh, Nov 27 2012
a(n) is the sum of the positive differences of the partition parts of 2n + 2 into exactly two parts (see example). - Wesley Ivan Hurt, Jun 02 2013
a(n)/a(n-1) is asymptotic to e^(2/n). - Richard R. Forberg, Jun 22 2013
Number of positive roots in the root system of type D_{n + 1} (for n > 2). - Tom Edgar, Nov 05 2013
Number of roots in the root system of type A_n (for n > 0). - Tom Edgar, Nov 05 2013
From Felix P. Muga II, Mar 18 2014: (Start)
a(m), for m >= 1, are the only positive integer values t for which the Binet-de Moivre formula for the recurrence b(n) = b(n-1) + t*b(n-2) with b(0) = 0 and b(1) = 1 has a root of a square. PROOF (as suggested by Wolfdieter Lang, Mar 26 2014): The sqrt(1 + 4t) appearing in the zeros r1 and r2 of the characteristic equation is (a positive) integer for positive integer t precisely if 4t + 1 = (2m + 1)^2, that is t = a(m), m >= 1. Thus, the characteristic roots are integers: r1 = m + 1 and r2 = -m.
Let m > 1 be an integer. If b(n) = b(n-1) + a(m)*b(n-2), n >= 2, b(0) = 0, b(1) = 1, then lim_{n->oo} b(n+1)/b(n) = m + 1. (End)
Cf. A130534 for relations to colored forests, disposition of flags on flagpoles, and colorings of the vertices (chromatic polynomial) of the complete graphs (here simply K_2). - Tom Copeland, Apr 05 2014
The set of integers k for which k + sqrt(k + sqrt(k + sqrt(k + sqrt(k + ...) ... is an integer. - Leslie Koller, Apr 11 2014
a(n-1) is the largest number k such that (n*k)/(n+k) is an integer. - Derek Orr, May 22 2014
Number of ways to place a domino and a singleton on a strip of length n - 2. - Ralf Stephan, Jun 09 2014
With offset 1, this appears to give the maximal number of crossings between n nonconcentric circles of equal radius. - Felix Fröhlich, Jul 14 2014
For n > 1, the harmonic mean of the n values a(1) to a(n) is n + 1. The lowest infinite sequence of increasing positive integers whose cumulative harmonic mean is integral. - Ian Duff, Feb 01 2015
a(n) is the maximum number of queens of one color that can coexist without attacking one queen of the opponent's color on an (n+2) X (n+2) chessboard. The lone queen can be placed in any position on the perimeter of the board. - Bob Selcoe, Feb 07 2015
With a(0) = 1, a(n-1) is the smallest positive number not in the sequence such that Sum_{i = 1..n} 1/a(i-1) has a denominator equal to n. - Derek Orr, Jun 17 2015
The positive members of this sequence are a proper subsequence of the so-called 1-happy couple products A007969. See the W. Lang link there, eq. (4), with Y_0 = 1, with a table at the end. - Wolfdieter Lang, Sep 19 2015
For n > 0, a(n) is the reciprocal of the area bounded above by y = x^(n-1) and below by y = x^n for x in the interval [0, 1]. Summing all such areas visually demonstrates the formula below giving Sum_{n >= 1} 1/a(n) = 1. - Rick L. Shepherd, Oct 26 2015
It appears that, except for a(0) = 0, this is the set of positive integers n such that x*floor(x) = n has no solution. (For example, to get 3, take x = -3/2.) - Melvin Peralta, Apr 14 2016
If two independent real random variables, x and y, are distributed according to the same exponential distribution: pdf(x) = lambda * exp(-lambda * x), lambda > 0, then the probability that n - 1 <= x/y < n is given by 1/a(n). - Andres Cicuttin, Dec 03 2016
a(n) is equal to the sum of all possible differences between n different pairs of consecutive odd numbers (see example). - Miquel Cerda, Dec 04 2016
a(n+1) is the dimension of the space of vector fields in the plane with polynomial coefficients up to order n. - Martin Licht, Dec 04 2016
It appears that a(n) + 3 is the area of the largest possible pond in a square (A268311). - Craig Knecht, May 04 2017
Also the number of 3-cycles in the (n+3)-triangular honeycomb acute knight graph. - Eric W. Weisstein, Jul 27 2017
Also the Wiener index of the (n+2)-wheel graph. - Eric W. Weisstein, Sep 08 2017
The left edge of a Floyd's triangle that consists of even numbers: 0; 2, 4; 6, 8, 10; 12, 14, 16, 18; 20, 22, 24, 26, 28; ... giving 0, 2, 6, 12, 20, ... The right edge generates A028552. - Waldemar Puszkarz, Feb 02 2018
a(n+1) is the order of rowmotion on a poset obtained by adjoining a unique minimal (or maximal) element to a disjoint union of at least two chains of n elements. - Nick Mayers, Jun 01 2018
From Juhani Heino, Feb 05 2019: (Start)
For n > 0, 1/a(n) = n/(n+1) - (n-1)/n.
For example, 1/6 = 2/3 - 1/2; 1/12 = 3/4 - 2/3.
Corollary of this:
Take 1/2 pill.
Next day, take 1/6 pill. 1/2 + 1/6 = 2/3, so your daily average is 1/3.
Next day, take 1/12 pill. 2/3 + 1/12 = 3/4, so your daily average is 1/4.
And so on. (End)
From Bernard Schott, May 22 2020: (Start)
For an oblong number m >= 6 there exists a Euclidean division m = d*q + r with q < r < d which are in geometric progression, in this order, with a common integer ratio b. For b >= 2 and q >= 1, the Euclidean division is m = qb*(qb+1) = qb^2 * q + qb where (q, qb, qb^2) are in geometric progression.
Some examples with distinct ratios and quotients:
   6 | 4            30 | 25          42 | 18
     -----             -----            -----
   2 | 1      ,      5 |  1   ,       6 |  2    ,
and also:
  42 | 12          420 | 100
     -----             -----
   6 |  3     ,     20 |   4  .
Some oblong numbers also satisfy a Euclidean division m = d*q + r with q < r < d that are in geometric progression in this order but with a common noninteger ratio b > 1 (see A335064). (End)
For n >= 1, the continued fraction expansion of sqrt(a(n)) is [n; {2, 2n}]. For n=1, this collapses to [1; {2}]. - Magus K. Chu, Sep 09 2022
a(n-2) is the maximum irregularity over all trees with n vertices. The extremal graphs are stars. (The irregularity of a graph is the sum of the differences between the degrees over all edges of the graph.) - Allan Bickle, May 29 2023
For n > 0, number of diagonals in a regular 2*(n+1)-gon that are not parallel to any edge (cf. A367204). - Paolo Xausa, Mar 30 2024
a(n-1) is the maximum Zagreb index over all trees with n vertices. The extremal graphs are stars. (The Zagreb index of a graph is the sum of the squares of the degrees over all vertices of the graph.) - Allan Bickle, Apr 11 2024
For n >= 1, a(n) is the determinant of the distance matrix of a cycle graph on 2*n + 1 vertices (if the length of the cycle is even such a determinant is zero). - Miquel A. Fiol, Aug 20 2024
For n > 1, the continued fraction expansion of sqrt(16*a(n)) is [2n+1; {1, 2n-1, 1, 8n+2}]. - Magus K. Chu, Nov 20 2024
For n>=2, a(n) is the number of faces on a n+1-zone rhombic zonohedron. Each pair of a collection of great circles on a sphere intersects at two points, so there are 2*binomial(n+1,2) intersections. The dual of the implied polyhedron is a rhombic zonohedron, its faces corresponding to the intersections. - Shel Kaphan, Aug 12 2025

			a(3) = 12, since 2(3)+2 = 8 has 4 partitions with exactly two parts: (7,1), (6,2), (5,3), (4,4). Taking the positive differences of the parts in each partition and adding, we get: 6 + 4 + 2 + 0 = 12. - _Wesley Ivan Hurt_, Jun 02 2013
G.f. = 2*x + 6*x^2 + 12*x^3 + 20*x^4 + 30*x^5 + 42*x^6 + 56*x^7 + ... - _Michael Somos_, May 22 2014
From _Miquel Cerda_, Dec 04 2016: (Start)
a(1) = 2, since 45-43 = 2;
a(2) = 6, since 47-45 = 2 and 47-43 = 4, then 2+4 = 6;
a(3) = 12, since 49-47 = 2, 49-45 = 4, and 49-43 = 6, then 2+4+6 = 12. (End)

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James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 2-6.

T. D. Noe, Table of n, a(n) for n = 0..1000
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Alin Bostan, Frédéric Chyzak, and Vincent Pilaud, Refined product formulas for Tamari intervals, arXiv:2303.10986 [math.CO], 2023.
H. Bottomley, Illustration of initial terms of A000217, A002378.
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Index entries for "core" sequences
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Partial sums of A005843 (even numbers). Twice triangular numbers (A000217).
1/beta(n, 2) in A061928.
Cf. A035106, A087811, A119462, A127235, A049598, A124080, A033996, A028896, A046092, A000217, A005563, A046092, A001082, A059300, A059297, A059298, A166373, A002943 (bisection), A002939 (bisection), A078358 (complement).
A036689 and A036690 are subsequences. Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488. - Bruno Berselli, Jun 10 2013
Row n=2 of A185651.
Cf. A007745, A169810, A213541, A005369 (characteristic function).
Cf. A281026. - Bruno Berselli, Jan 16 2017
Cf. A045943 (4-cycles in triangular honeycomb acute knight graph), A028896 (5-cycles), A152773 (6-cycles).
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
A335064 is a subsequence.
Second column of A003506.
Cf. A002378, A046092, A028896 (irregularities of maximal k-degenerate graphs).
Cf. A347213 (Dgf at s=4).
Cf. A002378, A152811, A371912 (Zagreb indices of maximal k-degenerate graphs).
Cf. A016742, A049450.

a002378 n = n * (n + 1)
a002378_list = zipWith (*) [0..] [1..]
-- Reinhard Zumkeller, Aug 27 2012, Oct 12 2011

[n*(n+1) : n in [0..100]]; // Wesley Ivan Hurt, Oct 26 2015

A002378 := proc(n)
        n*(n+1) ;
end proc:
seq(A002378(n),n=0..100) ;

Table[n(n + 1), {n, 0, 50}] (* Robert G. Wilson v, Jun 19 2004 *)
oblongQ[n_] := IntegerQ @ Sqrt[4 n + 1]; Select[Range[0, 2600], oblongQ] (* Robert G. Wilson v, Sep 29 2011 *)
2 Accumulate[Range[0, 50]] (* Harvey P. Dale, Nov 11 2011 *)
LinearRecurrence[{3, -3, 1}, {2, 6, 12}, {0, 20}] (* Eric W. Weisstein, Jul 27 2017 *)

PARI
```
{a(n) = n*(n+1)};
```

concat(0, Vec(2*x/(1-x)^3 + O(x^100))) \\ Altug Alkan, Oct 26 2015

is(n)=my(m=sqrtint(n)); m*(m+1)==n \\ Charles R Greathouse IV, Nov 01 2018

is(n)=issquare(4*n+1) \\ Charles R Greathouse IV, Mar 16 2022

def a(n): return n*(n+1)
print([a(n) for n in range(51)]) # Michael S. Branicky, Jan 13 2022

(2 to 100 by 2).scanLeft(0)( + ) // Alonso del Arte, Sep 12 2019

G.f.: 2*x/(1-x)^3. - Simon Plouffe in his 1992 dissertation.
a(n) = a(n-1) + 2*n, a(0) = 0.
Sum_{n >= 1} a(n) = n*(n+1)*(n+2)/3 (cf. A007290, partial sums).
Sum_{n >= 1} 1/a(n) = 1. (Cf. Tijdeman)
Sum_{n >= 1} (-1)^(n+1)/a(n) = log(4) - 1 = A016627 - 1 [Jolley eq (235)].
1 = 1/2 + Sum_{n >= 1} 1/(2*a(n)) = 1/2 + 1/4 + 1/12 + 1/24 + 1/40 + 1/60 + ... with partial sums: 1/2, 3/4, 5/6, 7/8, 9/10, 11/12, 13/14, ... - Gary W. Adamson, Jun 16 2003
a(n)*a(n+1) = a(n*(n+2)); e.g., a(3)*a(4) = 12*20 = 240 = a(3*5). - Charlie Marion, Dec 29 2003
Sum_{k = 1..n} 1/a(k) = n/(n+1). - Robert G. Wilson v, Feb 04 2005
a(n) = A046092(n)/2. - Zerinvary Lajos, Jan 08 2006
Log 2 = Sum_{n >= 0} 1/a(2n+1) = 1/2 + 1/12 + 1/30 + 1/56 + 1/90 + ... = (1 - 1/2) + (1/3 - 1/4) + (1/5 - 1/6) + (1/7 - 1/8) + ... = Sum_{n >= 0} (-1)^n/(n+1) = A002162. - Gary W. Adamson, Jun 22 2003
a(n) = A110660(2*n). - N. J. A. Sloane, Sep 21 2005
a(n-1) = n^2 - n = A000290(n) - A000027(n) for n >= 1. a(n) is the inverse (frequency distribution) sequence of A000194(n). - Mohammad K. Azarian, Jul 26 2007
(2, 6, 12, 20, 30, ...) = binomial transform of (2, 4, 2). - Gary W. Adamson, Nov 28 2007
a(n) = 2*Sum_{i=0..n} i = 2*A000217(n). - Artur Jasinski, Jan 09 2007, and Omar E. Pol, May 14 2008
a(n) = A006503(n) - A000292(n). - Reinhard Zumkeller, Sep 24 2008
a(n) = A061037(4*n) = (n+1/2)^2 - 1/4 = ((2n+1)^2 - 1)/4 = (A005408(n)^2 - 1)/4. - Paul Curtz, Oct 03 2008 and Klaus Purath, Jan 13 2022
a(0) = 0, a(n) = a(n-1) + 1 + floor(x), where x is the minimal positive solution to fract(sqrt(a(n-1) + 1 + x)) = 1/2. - Hieronymus Fischer, Dec 31 2008
E.g.f.: (x+2)*x*exp(x). - Geoffrey Critzer, Feb 06 2009
Product_{i >= 2} (1-1/a(i)) = -2*sin(Pi*A001622)/Pi = -2*sin(A094886)/A000796 = 2*A146481. - R. J. Mathar, Mar 12 2009, Mar 15 2009
E.g.f.: ((-x+1)*log(-x+1)+x)/x^2 also Integral_{x = 0..1} ((-x+1)*log(-x+1) + x)/x^2 = zeta(2) - 1. - Stephen Crowley, Jul 11 2009
a(A007018(n)) = A007018(n+1), i.e., A007018(n+1) = A007018(n)-th oblong numbers. - Jaroslav Krizek, Sep 13 2009
a(n) = floor((n + 1/2)^2). a(n) = A035608(n) + A004526(n+1). - Reinhard Zumkeller, Jan 27 2010
a(n) = 2*(2*A006578(n) - A035608(n)). - Reinhard Zumkeller, Feb 07 2010
a(n-1) = floor(n^5/(n^3 + n^2 + 1)). - Gary Detlefs, Feb 11 2010
For n > 1: a(n) = A173333(n+1, n-1). - Reinhard Zumkeller, Feb 19 2010
a(n) = A004202(A000217(n)). - Reinhard Zumkeller, Feb 12 2011
a(n) = A188652(2*n+1) + 1. - Reinhard Zumkeller, Apr 13 2011
For n > 0 a(n) = 1/(Integral_{x=0..Pi/2} 2*(sin(x))^(2*n-1)*(cos(x))^3). - Francesco Daddi, Aug 02 2011
a(n) = A002061(n+1) - 1. - Omar E. Pol, Oct 03 2011
a(0) = 0, a(n) = A005408(A034856(n)) - A005408(n-1). - Ivan N. Ianakiev, Dec 06 2012
a(n) = A005408(A000096(n)) - A005408(n). - Ivan N. Ianakiev, Dec 07 2012
a(n) = A001318(n) + A085787(n). - Omar E. Pol, Jan 11 2013
Sum_{n >= 1} 1/(a(n))^(2s) = Sum_{t = 1..2*s} binomial(4*s - t - 1, 2*s - 1) * ( (1 + (-1)^t)*zeta(t) - 1). See Arxiv:1301.6293. - R. J. Mathar, Feb 03 2013
a(n)^2 + a(n+1)^2 = 2 * a((n+1)^2), for n > 0. - Ivan N. Ianakiev, Apr 08 2013
a(n) = floor(n^2 * e^(1/n)) and a(n-1) = floor(n^2 / e^(1/n)). - Richard R. Forberg, Jun 22 2013
a(n) = 2*C(n+1, 2), for n >= 0. - Felix P. Muga II, Mar 11 2014
A005369(a(n)) = 1. - Reinhard Zumkeller, Jul 05 2014
Binomial transform of [0, 2, 2, 0, 0, 0, ...]. - Alois P. Heinz, Mar 10 2015
a(2n) = A002943(n) for n >= 0, a(2n-1) = A002939(n) for n >= 1. - M. F. Hasler, Oct 11 2015
For n > 0, a(n) = 1/(Integral_{x=0..1} (x^(n-1) - x^n) dx). - Rick L. Shepherd, Oct 26 2015
a(n) = A005902(n) - A007588(n). - Peter M. Chema, Jan 09 2016
For n > 0, a(n) = lim_{m -> oo} (1/m)*1/(Sum_{i=m*n..m*(n+1)} 1/i^2), with error of ~1/m. - Richard R. Forberg, Jul 27 2016
From Ilya Gutkovskiy, Jul 28 2016: (Start)
Dirichlet g.f.: zeta(s-2) + zeta(s-1).
Convolution of nonnegative integers (A001477) and constant sequence (A007395).
Sum_{n >= 0} a(n)/n! = 3*exp(1). (End)
From Charlie Marion, Mar 06 2020: (Start)
a(n)*a(n+2k-1) + (n+k)^2 = ((2n+1)*k + n^2)^2.
a(n)*a(n+2k) + k^2 = ((2n+1)*k + a(n))^2. (End)
Product_{n>=1} (1 + 1/a(n)) = cosh(sqrt(3)*Pi/2)/Pi. - Amiram Eldar, Jan 20 2021
A generalization of the Dec 29 2003 formula, a(n)*a(n+1) = a(n*(n+2)), follows. a(n)*a(n+k) = a(n*(n+k+1)) + (k-1)*n*(n+k+1). - Charlie Marion, Jan 02 2023
a(n) = A016742(n) - A049450(n). - Leo Tavares, Mar 15 2025

Additional comments from Michael Somos

Comment and cross-reference added by Christopher Hunt Gribble, Oct 13 2009

A048993 Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 7, 6, 1, 0, 1, 15, 25, 10, 1, 0, 1, 31, 90, 65, 15, 1, 0, 1, 63, 301, 350, 140, 21, 1, 0, 1, 127, 966, 1701, 1050, 266, 28, 1, 0, 1, 255, 3025, 7770, 6951, 2646, 462, 36, 1, 0, 1, 511, 9330, 34105, 42525, 22827, 5880, 750, 45, 1

Offset: 0

N. J. A. Sloane, Dec 11 1999

Also known as Stirling set numbers.
S(n,k) enumerates partitions of an n-set into k nonempty subsets.
The o.g.f. for the sequence of diagonal k (k=0 for the main diagonal) is G(k,x) = ((x^k)/(1-x)^(2*k+1))*Sum_{m=0..k-1} A008517(k,m+1)*x^m. A008517 is the second-order Eulerian triangle. - Wolfdieter Lang, Oct 14 2005
From Philippe Deléham, Nov 14 2007: (Start)
Sum_{k=0..n} S(n,k)*x^k = B_n(x), where B_n(x) = Bell polynomials.
The first few Bell polynomials are:
  B_0(x) = 1;
  B_1(x) = 0 + x;
  B_2(x) = 0 + x +  x^2;
  B_3(x) = 0 + x +  3x^2 +   x^3;
  B_4(x) = 0 + x +  7x^2 +  6x^3 +   x^4;
  B_5(x) = 0 + x + 15x^2 + 25x^3 + 10x^4 +   x^5;
  B_6(x) = 0 + x + 31x^2 + 90x^3 + 65x^4 + 15x^5 + x^6;
(End)
This is the Sheffer triangle (1, exp(x) - 1), an exponential (binomial) convolution triangle. The a-sequence is given by A006232/A006233 (Cauchy sequence). The z-sequence is the zero sequence. See the link under A006232 for the definition and use of these sequences. The row sums give A000110 (Bell), and the alternating row sums give A000587 (see the Philippe Deléham formulas and crossreferences below). - Wolfdieter Lang, Oct 16 2014
Also the inverse Bell transform of the factorial numbers (A000142). For the definition of the Bell transform see A264428 and for cross-references A265604. - Peter Luschny, Dec 31 2015
From Wolfdieter Lang, Feb 21 2017: (Start)
The transposed (trans) of this lower triagonal Sheffer matrix of the associated type S = (1, exp(x)  - 1) (taken as N X N matrix for arbitrarily large N) provides the transition matrix from the basis {x^n/n!}, n >= 0, to the basis {y^n/n!}, n >= 0, with y^n/n! = Sum_{m>=n} S^{trans}(n, m) x^m/m! = Sum_{m>=0} x^m/m!*S(m, n).
The Sheffer transform with S = (g, f) of a sequence {a_n} to {b_n} for n >= 0, in matrix notation  vec(b) = S vec(a), satisfies, with e.g.f.s A and B, B(x) = g(x)*A(f(x)) and  B(x) = A(y(x)) identically, with vec(xhat) =  S^{trans,-1} vec(yhat) in symbolic notation with vec(xhat)_n = x^n/n! (similarly for vec(yhat)).
(End)
For k >= 1 S(n, k) = h^{(k)}{n-k}, the complete homogeneous symmetric function of the k symbols 1,2, ..., k, of degree n-k. Thus S(n, k) is for k >= 1  the (dimensionless) volume of the multichoose(k, n-k) = binomial(n-1, k-1) polytopes of dimension n-k with (dimensionless) side lengths from the set {1, 2, ..., k}. See an example below. - _Wolfdieter Lang, May 26 2017
Number of partitions of {1, 2, ..., n+1} into k+1 nonempty subsets such that no subset contains two adjacent numbers. - Thomas Anton, Sep 26 2022

			The triangle S(n,k) begins:
  n\k 0 1    2     3      4       5       6      7      8     9   10 11 12
  0:  1
  1:  0 1
  2:  0 1    1
  3:  0 1    3     1
  4:  0 1    7     6      1
  5:  0 1   15    25     10       1
  6:  0 1   31    90     65      15       1
  7:  0 1   63   301    350     140      21      1
  8:  0 1  127   966   1701    1050     266     28      1
  9:  0 1  255  3025   7770    6951    2646    462     36     1
 10:  0 1  511  9330  34105   42525   22827   5880    750    45    1
 11:  0 1 1023 28501 145750  246730  179487  63987  11880  1155   55  1
 12:  0 1 2047 86526 611501 1379400 1323652 627396 159027 22275 1705 66  1
 ... reformatted and extended - _Wolfdieter Lang_, Oct 16 2014
Completely symmetric function S(4, 2) = h^{(2)}_2 = 1^2 + 2^2 + 1^1*2^1 = 7; S(5, 2) = h^{(2)}_3 = 1^3 + 2^3 + 1^2*2^1 + 1^1*2^2 = 15. - _Wolfdieter Lang_, May 26 2017
From _Wolfdieter Lang_, Aug 11 2017: (Start)
Recurrence: S(5, 3) = S(4, 2) + 2*S(4, 3) = 7 + 3*6 = 25.
Boas-Buck recurrence for column m = 3, and n = 5: S(5, 3) = (3/2)*((5/2)*S(4, 3) + 10*Bernoulli(2)*S(3, 3)) = (3/2)*(15 + 10*(1/6)*1) = 25. (End)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 835.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 310.
J. H. Conway and R. K. Guy, The Book of Numbers, Springer, p. 92.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 244.
J. Riordan, An Introduction to Combinatorial Analysis, p. 48.

David W. Wilson, Table of n, a(n) for n = 0..10010
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
V. E. Adler, Set partitions and integrable hierarchies, arXiv:1510.02900 [nlin.SI], 2015.
Peter Bala, The white diamond product of power series
Paul Barry, Generalized Stirling Numbers, Exponential Riordan Arrays, and Toda Chain Equations, Journal of Integer Sequences, 17 (2014), #14.2.3.
Paul Barry, Constructing Exponential Riordan Arrays from Their A and Z Sequences, Journal of Integer Sequences, 17 (2014), #14.2.6.
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Xi Chen, Bishal Deb, Alexander Dyachenko, Tomack Gilmore, and Alan D. Sokal, Coefficientwise total positivity of some matrices defined by linear recurrences, arXiv:2012.03629 [math.CO], 2020.
R. M. Dickau, Stirling numbers of the second kind
Gerard Duchamp, Karol A. Penson, Allan I. Solomon, Andrej Horzela, and Pawel Blasiak, One-parameter groups and combinatorial physics, arXiv:quant-ph/0401126, 2004.
FindStat - Combinatorial Statistic Finder, The number of blocks in the set partition.
Bill Gosper, Colored illustrations of triangle of Stirling numbers of second kind read mod 2, 3, 4, 5, 6, 7
W. Steven Gray and Makhin Thitsa, System Interconnections and Combinatorial Integer Sequences, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-11 March 2013, Digital Object Identifier: 10.1109/SSST.2013.6524939.
Aoife Hennessy and Paul Barry, Generalized Stirling Numbers, Exponential Riordan Arrays, and Orthogonal Polynomials, J. Int. Seq. 14 (2011) # 11.8.2.
Paweł Hitczenko, A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality, arXiv:2403.03422 [math.CO], 2024. See pp. 8-9.
Mathias Pétréolle and Alan D. Sokal, Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions, arXiv:1907.02645 [math.CO], 2019.
Claus Michael Ringel, The Catalan combinatorics of the hereditary artin algebras, arXiv preprint arXiv:1502.06553 [math.RT], 2015.
X.-T. Su, D.-Y. Yang, and W.-W. Zhang, A note on the generalized factorial, Australasian Journal of Combinatorics, Volume 56 (2013), Pages 133-137.

See especially A008277 which is the main entry for this triangle.
Cf. A008275, A039810-A039813, A048994.
A000110(n) = sum(S(n, k)) k=0..n, n >= 0. Cf. A085693.
Cf. A084938, A106800 (mirror image), A138378, A213061 (mod 2).
Cf. A059297, A354794.

a048993 n k = a048993_tabl !! n !! k
a048993_row n = a048993_tabl !! n
a048993_tabl = iterate (\row ->
   [0] ++ (zipWith (+) row $ zipWith (*) [1..] $ tail row) ++ [1]) [1]
-- Reinhard Zumkeller, Mar 26 2012

for n from 0 to 10 do seq(Stirling2(n,k),k=0..n) od; # yields sequence in triangular form # Emeric Deutsch, Nov 01 2006

t[n_, k_] := StirlingS2[n, k]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Robert G. Wilson v *)

create_list(stirling2(n,k),n,0,12,k,0,n); /* Emanuele Munarini, Mar 11 2011 */

for(n=0, 22, for(k=0, n, print1(stirling(n, k, 2), ", ")); print()); \\ Joerg Arndt, Apr 21 2013

S(n, k) = k*S(n-1, k) + S(n-1, k-1), n > 0; S(0, k) = 0, k > 0; S(0, 0) = 1.
Equals [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is Deléham's operator defined in A084938.
Sum_{k = 0..n} x^k*S(n, k) = A213170(n), A000587(n), A000007(n), A000110(n), A001861(n), A027710(n), A078944(n), A144180(n), A144223(n), A144263(n) respectively for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7. - Philippe Deléham, May 09 2004, Feb 16 2013
S(n, k) = Sum_{i=0..k} (-1)^(k+i)binomial(k, i)i^n/k!. - Paul Barry, Aug 05 2004
Sum_{k=0..n} k*S(n,k) = B(n+1)-B(n), where B(q) are the Bell numbers (A000110). - Emeric Deutsch, Nov 01 2006
Equals the inverse binomial transform of A008277. - Gary W. Adamson, Jan 29 2008
G.f.: 1/(1-xy/(1-x/(1-xy/(1-2x/(1-xy/1-3x/(1-xy/(1-4x/(1-xy/(1-5x/(1-... (continued fraction equivalent to Deléham DELTA construction). - Paul Barry, Dec 06 2009
G.f.: 1/Q(0), where Q(k) = 1 - (y+k)*x - (k+1)*y*x^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 09 2013
Inverse of padded A008275 (padded just as A048993 = padded A008277). - Tom Copeland, Apr 25 2014
E.g.f. for the row polynomials s(n,x) = Sum_{k=0..n} S(n,k)*x^k is exp(x*(exp(z)-1)) (Sheffer property). E.g.f. for the k-th column sequence with k leading zeros is ((exp(x)-1)^k)/k! (Sheffer property). - Wolfdieter Lang, Oct 16 2014
G.f. for column k: x^k/Product_{j=1..k} (1-j*x), k >= 0 (with the empty product for k = 0 put to 1). See Abramowitz-Stegun, p. 824, 24.1.4 B. - Wolfdieter Lang, May 26 2017
Boas-Buck recurrence for column sequence m: S(n, k) = (k/(n - k))*(n*S(n-1, k)/2 + Sum_{p=k..n-2} (-1)^(n-p)*binomial(n,p)*Bernoulli(n-p)*S(p, k)), for n > k >= 0, with input T(k,k) = 1. See a comment and references in A282629. An example is given below. - Wolfdieter Lang, Aug 11 2017
The n-th row polynomial has the form x o x o ... o x (n factors), where o denotes the white diamond multiplication operator defined in Bala - see Example E4. - Peter Bala, Jan 07 2018
Sum_{k=1..n} k*S(n,k) = A138378(n). - Alois P. Heinz, Jan 07 2022
S(n,k) = Sum_{j=k..n} (-1)^(j-k)*A059297(n,j)*A354794(j,k). - Mélika Tebni, Jan 27 2023

A088956 Triangle, read by rows, of coefficients of the hyperbinomial transform.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 16, 9, 3, 1, 125, 64, 18, 4, 1, 1296, 625, 160, 30, 5, 1, 16807, 7776, 1875, 320, 45, 6, 1, 262144, 117649, 27216, 4375, 560, 63, 7, 1, 4782969, 2097152, 470596, 72576, 8750, 896, 84, 8, 1, 100000000, 43046721, 9437184, 1411788, 163296, 15750

Offset: 0

Paul D. Hanna, Oct 26 2003

The hyperbinomial transform of a sequence {b} is defined to be the sequence {d} given by d(n) = Sum_{k=0..n} T(n,k)*b(k), where T(n,k) = (n-k+1)^(n-k-1)*C(n,k).
Given a table in which the n-th row is the n-th binomial transform of the first row, then the hyperbinomial transform of any diagonal results in the next lower diagonal in the table.
The simplest example of a table of iterated binomial transforms is A009998, with a main diagonal of {1,2,9,64,625,...}; and the hyperbinomial transform of this diagonal gives the next lower diagonal, {1,3,16,125,1296,...}, since 1=(1)*1, 3=(1)*1+(1)*2, 16=(3)*1+(2)*2+(1)*9, 125=(16)*1+(9)*2+(3)*9+(1)*64, etc.
Another example: the hyperbinomial transform maps A065440 into A055541, since HYPERBINOMIAL([1,1,1,8,81,1024,15625]) = [1,2,6,36,320,3750,54432] where e.g.f.: A065440(x)+x = x-x/( LambertW(-x)*(1+LambertW(-x)) ), e.g.f.: A055541(x) = x-x*LambertW(-x).
The m-th iteration of the hyperbinomial transform is given by the triangle of coefficients defined by T_m(n,k) = m*(n-k+m)^(n-k-1)*binomial(n,k).
Example: PARI code for T_m: {a=[1,1,1,8,81,1024,15625]; m=1; b=vector(length(a)); for(n=0,length(a)-1, b[n+1]=sum(k=0,n, m*(n-k+m)^(n-k-1)*binomial(n,k)*a[k+1]); print1(b[n+1],","))} RETURNS b=[1,2,6,36,320,3750,54432].
The INVERSE hyperbinomial transform is thus given by m=-1: {a=[1,2,6,36,320,3750,54432]; m=-1; b=vector(length(a)); for(n=0,length(a)-1, b[n+1]=sum(k=0,n, m*(n-k+m)^(n-k-1)*binomial(n,k)*a[k+1]); print1(b[n+1],","))} RETURNS b=[1,1,1,8,81,1024,15625].
Simply stated, the HYPERBINOMIAL transform is to -LambertW(-x)/x as the BINOMIAL transform is to exp(x).
Let A[n] be the set of all forests of labeled rooted trees on n nodes. Build a superset B[n] of A[n] by designating "some" (possibly all or none) of the isolated nodes in each forest. T(n,k) is the number of elements in B[n] with exactly k designated nodes. See A219034. - Geoffrey Critzer, Nov 10 2012

			Rows begin:
       {1},
       {1,      1},
       {3,      2,     1},
      {16,      9,     3,    1},
     {125,     64,    18,    4,   1},
    {1296,    625,   160,   30,   5,  1},
   {16807,   7776,  1875,  320,  45,  6, 1},
  {262144, 117649, 27216, 4375, 560, 63, 7, 1}, ...

T. D. Noe, Rows n=0..50 of triangle, flattened
T. Copeland, Composition, Conjugation, and the Umbral Calculus-Part I, 2021.
G. Helms, Pascalmatrix tetrated
E. W. Weisstein, Abel Polynomial, From MathWorld - A Wolfram Web Resource.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A088957 (row sums), A000272 (first column), A009998, A105599, A132440, A215534 (matrix inverse), A215652.
Cf. A227325 (central terms).
Cf. A007318, A059297, A137542.

a088956 n k =  a095890 (n + 1) (k + 1) * a007318' n k `div` (n - k + 1)
a088956_row n = map (a088956 n) [0..n]
a088956_tabl = map a088956_row [0..]
-- Reinhard Zumkeller, Jul 07 2013

nn=8; t=Sum[n^(n-1)x^n/n!, {n,1,nn}]; Range[0,nn]! CoefficientList[Series[Exp[t+y x] ,{x,0,nn}], {x,y}] //Grid (* Geoffrey Critzer, Nov 10 2012 *)

T(n, k) = (n-k+1)^(n-k-1)*C(n, k).
E.g.f.: -LambertW(-x)*exp(x*y)/x. - Vladeta Jovovic, Oct 27 2003
From Peter Bala, Sep 11 2012: (Start)
Let T(x) = Sum_{n >= 0} n^(n-1)*x^n/n! denote the tree function of A000169. The e.g.f. is (T(x)/x)*exp(t*x) = exp(T(x))*exp(t*x) = 1 + (1 + t)*x + (3 + 2*t + t^2)*x^2/2! + .... Hence the triangle is the exponential Riordan array [T(x)/x,x] belonging to the exponential Appell group.
The matrix power (A088956)^r has the e.g.f. exp(r*T(x))*exp(t*x) with triangle entries given by r*(n-k+r)^(n-k-1)*binomial(n,k) for n and k >= 0. See A215534 for the case r = -1.
Let A(n,x) = x*(x+n)^(n-1) be an Abel polynomial. The present triangle is the triangle of connection constants expressing A(n,x+1) as a linear combination of the basis polynomials A(k,x), 0 <= k <= n. For example, A(4,x+1) = 125*A(0,x) + 64*A(1,x) + 18*A(2,x) + 4*A(3,x) + A(4,x) gives row 4 as [125,64,18,4,1].
Let S be the array with the sequence [1,2,3,...] on the main subdiagonal and zeros elsewhere. S is the infinitesimal generator for Pascal's triangle (see A132440). Then the infinitesimal generator for this triangle is S*A088956; that is, A088956 = Exp(S*A088956), where Exp is the matrix exponential.
With T(x) the tree function as above, define E(x) = T(x)/x. Then A088956 = E(S) = Sum_{n>=0} (n+1)^(n-1)*S^n/n!.
For commuting lower unit triangular matrices A and B, we define A raised to the matrix power B, denoted A^^B, to be the matrix Exp(B*log(A)), where the matrix logarithm Log(A) is defined as Sum_{n >= 1} (-1)^(n+1)*(A-1)^n/n. Let P denote Pascal's triangle A007318. Then the present triangle, call it X, solves the matrix equation P^^X = X . See A215652 for the solution to X^^P = P. Furthermore, if we denote the inverse of X by Y then X^^Y = P. As an infinite tower of matrix powers, A088956 = P^^(P^^(P^^(...))).
A088956 augmented with the sequence (x,x,x,...) on the first superdiagonal is the production matrix for the row polynomials of A105599.
(End)
T(n,k) = A095890(n+1,k+1) * A007318(n,k) / (n-k+1), 0 <= k <= n. - Reinhard Zumkeller, Jul 07 2013
Sum_{k = 0..n} T(n,n-k)*(x - k - 1)^(n-k) = x^n. Setting x = n + 1 gives Sum_{k = 0..n} T(n,k)*k^k = (n + 1)^n. - Peter Bala, Feb 17 2017
As lower triangular matrices, this entry, T, equals unsigned A137542 * A007318 * signed A059297. The Pascal matrix is sandwiched between a pair of inverse matrices, so this entry is conjugate to the Pascal matrix, allowing convergent analytic expressions of T, say f(T), to be computed as f(A007318) sandwiched between the inverse pair. - Tom Copeland, Dec 06 2021

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

If a triangular array has an e.g.f. of the form exp(t*F(x)) with F(0) = 0, then the o.g.f.'s for the diagonals of the triangle are rational functions in t [Drake, 1.10]. The rational functions are the coefficients in the compositional inverse (with respect to x) (x-t*F(x))^(-1).

Triangle A059297 has e.g.f. exp(t*x*exp(x)). The present triangle lists the coefficients of the numerator polynomials of the rational o.g.f.'s of the diagonals of A059297. Drake, Example 1.10.9, gives three combinatorial interpretations for these coefficients (but note the expansion at the bottom of p.68 is for (x-t*(-W(-x))^(-1), W(x) the Lambert W function, and not for (x-t*x*exp(x))^(-1) as stated there). Row reversal of A155163.

Row sums give A177885.
The Abel polynomials are associated with the Abel operator t*exp(y*t)*p(x) = t*p(x+y).
From Peter Luschny, Jan 14 2009: (Start)
Abs(T(n,k)) is the number of rooted labeled trees on n+1 vertices with a root degree k (Clarke's formula).
The row sums in the unsigned case, Sum_{k=0..n} abs(T(n,k)), count the trees on n+1 labeled nodes, A000272(n+1). (End)
Exponential Riordan array [1, W(x)], W(x) the Lambert W-function. - Paul Barry, Nov 19 2010
The inverse array is the exponential Riordan array [1, x*exp(x)], which is A059297. - Peter Bala, Apr 08 2013
The inverse Bell transform of [1,2,3,...]. See A264428 for the Bell transform and A264429 for the inverse Bell transform. - Peter Luschny, Dec 20 2015
Also the Bell transform of (-1)^n*(n+1)^n. - Peter Luschny, Jan 18 2016

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) = 1 for all n. The case S(n) = n gives the triangle of idempotent numbers A059297.
From Manfred Boergens, Mar 04 2025: (Start)
T(n,k) = number of collections of k+1 disjoint [n]-subsets covering [n], with [0]={}.
For disjoint covers (collections without an empty set) see A008277.
For non-disjoint collections see A163353.
For non-disjoint covers see A055154. (End)

With a different offset, number of n-permutations of 9 objects:
p, r, s, t, u, v, z, x, y with repetition allowed, containing exactly eight (8) u's. Example: a(1)=72 because we have
uuuuuuuup, uuuuuuupu, uuuuuupuu, uuuuupuuu, uuuupuuuu, uuupuuuuu, uupuuuuuu, upuuuuuuu, puuuuuuuu,
uuuuuuuur, uuuuuuuru, uuuuuuruu, uuuuuruuu, uuuuruuuu, uuuruuuuu, uuruuuuuu, uruuuuuuu, ruuuuuuuu,
uuuuuuuus, uuuuuuusu, uuuuuusuu, uuuuusuuu, uuuusuuuu, uuusuuuuu, uusuuuuuu, usuuuuuuu, suuuuuuuu,
uuuuuuuut, uuuuuuutu, uuuuuutuu, uuuuutuuu, uuuutuuuu, uuutuuuuu, uutuuuuuu, utuuuuuuu, tuuuuuuuu,
uuuuuuuuv, uuuuuuuvu, uuuuuuvuu, uuuuuvuuu, uuuuvuuuu, uuuvuuuuu, uuvuuuuuu, uvuuuuuuu, vuuuuuuuu,
uuuuuuuuz, uuuuuuuzu, uuuuuuzuu, uuuuuzuuu, uuuuzuuuu, uuuzuuuuu, uuzuuuuuu, uzuuuuuuu, zuuuuuuuu,
uuuuuuuux, uuuuuuuxu, uuuuuuxuu, uuuuuxuuu, uuuuxuuuu, uuuxuuuuu, uuxuuuuuu, uxuuuuuuu, xuuuuuuuu,
uuuuuuuuy, uuuuuuuyu, uuuuuuyuu, uuuuuyuuu, uuuuyuuuu, uuuyuuuuu, uuyuuuuuu, uyuuuuuuu, yuuuuuuuu.

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A202017 Triangle of coefficients of the numerator polynomials of the rational o.g.f.'s of the diagonals of A059297.

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A048993 Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.

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A088956 Triangle, read by rows, of coefficients of the hyperbinomial transform.

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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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A137452 Triangular array of the coefficients of the sequence of Abel polynomials A(n,x) := x*(x-n)^(n-1).

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