A061356 -id:A061356 - cp's OEIS frontend

A139526 Triangle A061356 read right to left.

1, 1, 2, 1, 6, 9, 1, 12, 48, 64, 1, 20, 150, 500, 625, 1, 30, 360, 2160, 6480, 7776, 1, 42, 735, 6860, 36015, 100842, 117649, 1, 56, 1344, 17920, 143360, 688128, 1835008, 2097152, 1, 72, 2268, 40824, 459270, 3306744, 14880348, 38263752, 43046721, 1, 90, 3600, 84000, 1260000, 12600000, 84000000, 360000000, 900000000, 1000000000

Offset: 2

Alford Arnold, Apr 24 2008

Related to the two Appell sequences the Bernoulli polynomials B(n,x) and their umbral compositional inverses (cf. A074909) Up(n,x) = [(x+1)^(n+1)-x^(n+1)] / (n+1). With offset 0, the row polynomials of this entry P(n,x) = (Up(n,0))^(-n) * [x + Up(n,0)]^n = (n+1)^n * [x + 1/(n+1)]^n. Compare to the Abel polynomials of A061356, which are also an Appell sequence. - Tom Copeland, Nov 14 2014

			(1) times (1) = (1)
(1 1) * (1 2) = (1 2)
(1 2 1 ) * (1 3 9) = (1 6 9)
(1 3 3 1) * (1 4 16 64) = (1 12 48 64)
etc.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA. Second ed. 1994.
Peter D. Schumer (2004), Mathematical Journeys, page 168, Proposition 16.1 (c)

P. Bala, Fractional iteration of a series inversion operator
Wikipedia, Lambert W function

Cf. A000272 (row sums), A061356 (row reverse), A028421, A074909, A000169 (main diagonal), A251592, A260687.

A061356 := proc(n,k) binomial(n-2,k-1)*(n-1)^(n-k-1); end: A139526 := proc(n,k) A061356(n,n-k-1) ; end: for n from 2 to 14 do for k from 0 to n-2 do printf("%d,",A139526(n,k)) ; od: od: # R. J. Mathar, May 22 2008

T[n_, k_] := (n - 1)^k*Binomial[n - 2, n - k - 2];
Table[T[n, k], {n, 2, 11}, {k, 0, n - 2}] // Flatten (* Jean-François Alcover, Jun 13 2023 *)

for(n=2,12,forstep(k=n-1,1,-1,print1(binomial(n-2, k-1)*(n-1)^(n-k-1)","))) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 10 2008

E.g.f. (with offset 1) Sum_{n >= 1} (1 + n*t)^(n-1)*x^n/n! = x + (1 + 2*t)*x^2/2! + (1 + 6*t + 9*t^2)*x^3/3! + .... For properties of this function see Graham et al., equations 5.60, 5.61 and 7.71. The e.g.f. is the series reversion with respect to x of the function log(1 + x)/(1 + x)^t, which is the e.g.f. for a signed version of A028421. - Peter Bala, Jul 18 2013
From Peter Bala, Nov 16 2015: (Start)
E.g.f. with offset 0 and constant term 1: A(x,t) = ( Sum_{n >= 0} (n + 1)^(n-1)*t^n*x^n/n! )^(1/t). This is the generalized exponential series E_t(x) in the terminology of Graham et al., Section 5.4.
A(x,t)^m = 1 + Sum_{n >= 1} m*(m + n*t)^(n-1)*x^n/n!.
log(A(x,t)) = Sum_{n >= 1} (n*t)^(n-1)*x^n/n! = 1/t*T(t*x), where T(z) is Euler's tree function. See A000169.
A(x,t) = ( 1/x* Revert( x*exp(-x*t)) )^(1/t), where Revert is the series reversion operator with respect to x.
In the notation of the Bala link the e.g.f. is I^t(e^x), where I^t is a fractional series inversion operator. Cf. A251592, which has o.g.f. I^t(1 + x), and A260687, which has o.g.f. I^t(1/(1 - x)). (End)

More terms from R. J. Mathar and Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2008

A028421 Triangle read by rows: T(n, k) = (k+1)*A132393(n+1, k+1), for 0 <= k <= n.

Original entry on oeis.org

1, 1, 2, 2, 6, 3, 6, 22, 18, 4, 24, 100, 105, 40, 5, 120, 548, 675, 340, 75, 6, 720, 3528, 4872, 2940, 875, 126, 7, 5040, 26136, 39396, 27076, 9800, 1932, 196, 8, 40320, 219168, 354372, 269136, 112245, 27216, 3822, 288, 9

Offset: 0

Peter Wiggen (wiggen(AT)math.psu.edu)

Previous name was: Number triangle f(n, k) from n-th differences of the sequence {1/m^2}{m >= 1}, for n >= 0; the n-th difference sequence is {(-1)^n*n!*P(n, m)/D(n, m)^2}{m >= 1} where P(n, x) is the row polynomial P(n, x) = Sum_{k=0..n} f(n,k)*x^k and D(n, x) = x*(x+1)*...*(x+n).
From Johannes W. Meijer, Oct 07 2009: (Start)
The higher-order exponential integrals E(x,m,n) are defined in A163931 and the general formula of the asymptotic expansion of E(x,m,n) can be found in A163932.
We used the general formula and the asymptotic expansion of E(x,m=1,n), see A130534, to determine that E(x,m=2,n) ~ (exp(-x)/x^2)*(1 - (1+2*n)/x + (2 + 6*n + 3*n^2)/x^2 - (6 + 22*n + 18*n^2 + 4*n^3)/x^3 + ...) which can be verified with the EA(x,2,n) formula, see A163932. The coefficients in the denominators of this expansion lead to the sequence given above.
The asymptotic expansion of E(x,m=2,n) leads for n from one to ten to known sequences, see the cross-references. With these sequences one can form the triangles A165674 (left hand columns) and A093905 (right hand columns).
(End)
For connections to an operator relation between log(x) and x^n(d/dx)^n, see A238363. - Tom Copeland, Feb 28 2014
From Wolfdieter Lang, Nov 25 2018: (Start)
The signed triangle t(n, k) := (-1)^{n-k}*f(n, k) gives (n+1)*N(-1;n,x) = Sum_{k=0..n} t(n, k)*x^k, where N(-1;n,x) are the Narumi polynomials with parameter a = -1 (see the Weisstein link).
The members of the n-th difference sequence of the sequence {1/m^2}_{m>=1} mentioned above satisfies the recurrence delta(n, m) = delta(n-1, m+1) - delta(n-1, m), for n >= 1, m >= 1, with input delta(0, m) = 1/m^2. The solution is delta(n, m) = (n+1)!*N(-1;n,-m)/risefac(m, n+1)^2, with Narumi polynomials N(-1;n,x) and the rising factorials risefac(x, n+1) = D(n, x) = x*(x+1)*...*(x+n).
The above mentioned row polynomials P satisfy P(n, x) = (-1)^n*(n + 1)*N(-1;n,-x), for n >= 0. The recurrence is P(n, x) = (-x^2*P(n-1, x+1) + (n+x)^2*P(n-1, x))/n, for n >= 1, and P(0, x) = 1. (End)
The triangle is the exponential Riordan square (cf. A321620) of -log(1-x) with an additional main diagonal of zeros. - Peter Luschny, Jan 03 2019

			The triangle T(n, k) begins:
n\k       0        1        2        3        4       5       6      7     8   9 10
------------------------------------------------------------------------------------
0:        1
1:        1        2
2:        2        6        3
3:        6       22       18        4
4:       24      100      105       40        5
5:      120      548      675      340       75       6
6:      720     3528     4872     2940      875     126       7
7:     5040    26136    39396    27076     9800    1932     196      8
8:    40320   219168   354372   269136   112245   27216    3822    288     9
9:   362880  2053152  3518100  2894720  1346625  379638   66150   6960   405  10
10: 3628800 21257280 38260728 33638000 17084650 5412330 1104411 145200 11880 550 11
... - _Wolfdieter Lang_, Nov 23 2018

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Eric Weisstein's World of Mathematics, Narumi Polynomial [here for a = -1].

Row sums give A000254(n+1), n >= 0.
Cf. A132393 (unsigned Stirling1), A061356, A139526, A321620.
From Johannes W. Meijer, Oct 07 2009: (Start)
A000142, A052517, 3*A000399, 5*A000482 are the first four left hand columns; A000027, A002411 are the first two right hand columns.
The asymptotic expansion of E(x,m=2,n) leads to A000254 (n=1), A001705 (n=2), A001711 (n=3), A001716 (n=4), A001721 (n=5), A051524 (n=6), A051545 (n=7), A051560 (n=8), A051562 (n=9), A051564 (n=10), A093905 (triangle) and A165674 (triangle).
Cf. A163931 (E(x,m,n)), A130534 (m=1), A163932 (m=3), A163934 (m=4), A074246 (E(x,m=2,n+1)). (End)

A028421 := proc(n,k) (-1)^(n+k)*(k+1)*Stirling1(n+1,k+1) end:
seq(seq(A028421(n,k), k=0..n), n=0..8);
# Johannes W. Meijer, Oct 07 2009, Revised Sep 09 2012
egf := (1 - t)^(-x - 1)*(1 - x*log(1 - t)):
ser := series(egf, t, 16): coefft := n -> expand(coeff(ser,t,n)):
seq(seq(n!*coeff(coefft(n), x, k), k = 0..n), n = 0..8); # Peter Luschny, Jun 12 2022

f[n_, k_] = (k + 1) StirlingS1[n + 1, k + 1] // Abs; Flatten[Table[f[n, k], {n, 0, 9}, {k, 0, n}]][[1 ;; 47]] (* Jean-François Alcover, Jun 01 2011, after formula *)

# uses[riordan_square from A321620]
riordan_square(-ln(1 - x), 10, True) # Peter Luschny, Jan 03 2019

E.g.f.: d/dt(-log(1-t)/(1-t)^x). - Vladeta Jovovic, Oct 12 2003
The e.g.f. with offset 1: y = x + (1 + 2*t)*x^2/2! + (2 + 6*t + 3*t^2)*x^3/3! + ... has series reversion with respect to x equal to y - (1 + 2*t)*y^2/2! + (1 + 3*t)^2*y^3/3! - (1 + 4*t)^3*y^4/4! + .... This is an e.g.f. for a signed version of A139526. - Peter Bala, Jul 18 2013
Recurrence: T(n, k) = 0 if n < k; if k = 0 then T(0, 0) = 1 and T(n, 0) = n * T(n-1, 0) for n >= 1, otherwise T(n, k) = n*T(n-1, k) + ((k+1)/k)*T(n-1, k-1). From the unsigned Stirling1 recurrence. - Wolfdieter Lang, Nov 25 2018

Edited by Wolfdieter Lang, Nov 23 2018

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

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

Original entry on oeis.org

1, 0, 1, 0, -2, 1, 0, 9, -6, 1, 0, -64, 48, -12, 1, 0, 625, -500, 150, -20, 1, 0, -7776, 6480, -2160, 360, -30, 1, 0, 117649, -100842, 36015, -6860, 735, -42, 1, 0, -2097152, 1835008, -688128, 143360, -17920, 1344, -56, 1, 0, 43046721, -38263752, 14880348, -3306744, 459270, -40824, 2268, -72, 1

Offset: 0

Roger L. Bagula, Apr 18 2008

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

			Triangle begins:
  1;
  0,        1;
  0,       -2,       1;
  0,        9,      -6,       1;
  0,      -64,      48,     -12,      1;
  0,      625,    -500,     150,    -20,      1;
  0,    -7776,    6480,   -2160,    360,    -30,    1;
  0,   117649, -100842,   36015,  -6860,    735,  -42,   1;
  0, -2097152, 1835008, -688128, 143360, -17920, 1344, -56, 1;

Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 14 and 29

Seiichi Manyama, Rows n = 0..139, flattened
W. Y. Chen, A general bijective algorithm for trees, PNAS December 1, 1990 vol. 87 no. 24 9635-9639.
L. E. Clarke, On Cayley's formula for counting trees, J. London Math. Soc. 33 (1958), 471-475.
Péter L. Erdős and L. A. Székely, Applications of Antilexicographic Order. I., An Enumerative Theory of Trees, Adv. in Appl. Math. 10, (1989) 488-496.
Eric Weisstein's World of Mathematics, Abel Polynomial.
Wikipedia, Abel Polynomials.
Bao-Xuan Zhu, Total positivity from a generalized cycle index polynomial, arXiv:2006.14485 [math.CO], 2020.

Row sums A177885.
Cf. A000272, A061356, A059297 (inverse array), A264429.
Cf. A061354, A095996, A264234, A061355, A049444.

T := proc(n,k) if n = 0 and k = 0 then 1 else binomial(n-1,k-1)*(-n)^(n-k) fi end; seq(print(seq(T(n,k),k=0..n)),n=0..7); # Peter Luschny, Jan 14 2009
# The function BellMatrix is defined in A264428.
BellMatrix(n -> (-n-1)^n, 9); # Peter Luschny, Jan 27 2016

a0 = 1 a[x, 0] = 1; a[x, 1] = x; a[x_, n_] := x*(x - a0*n)^(n - 1); Table[Expand[a[x, n]], {n, 0, 10}]; a1 = Table[CoefficientList[a[x, n], x], {n, 0, 10}]; Flatten[a1]
(* Second program: *)
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[Function[n, (-n-1)^n], rows = 12];
Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

# uses[inverse_bell_transform from A264429]
def A137452_matrix(dim):
    nat = [n for n in (1..dim)]
    return inverse_bell_transform(dim, nat)
A137452_matrix(10) # Peter Luschny, Dec 20 2015

Row n gives the coefficients of the expansion of x*(x-n)^(n-1).
Abs(T(n,k)) = C(n-1,k-1)*n^(n-k). - Peter Luschny, Jan 14 2009
From Wolfdieter Lang, Nov 08 2022: (Start)
From the exponential Riordan (also Sheffer of Jabotinsky) type (1, LambertW) array (see comments).
E.g.f. of column sequence k, LambertW(x)^k/k!, for k >= 0.
E.g.f. of row polynomials P_n(y) = Sum_{k=0..n} T(n, k)*y^k: exp(y*LambertW(x)).
Recurrence for T: T(n, k) = 0 for n < k; T(n, 0) = 1 for n = 0 otherwise 0; T(n, k) = (n/k)*Sum_{j=0..n-k} binomial(k-1+j,k-1)*(-1)^j*T(n-1, k-1+j). (Jabotinsky type convolution triangle, the e.g.f.s for the a- and z-sequences are exp(-x), and 0. See the link in A006232.)
Recurrence for column k of T: T(n, k) = 0 for n < k, T(k, k) = 1, for k >= 0 otherwise T(n, k) = (n!*k/(n-k))*Sum_{j=k..n-1} (1/j!)*beta(n-1-j)*T(j, k), where beta(n) = A264234(n+1)/A095996(n+1) = {-1, 2, -9/2, 32/3, -625/24, ...} with o.g.f. d/dx(log(LambertW(x)/x)). See the Boas-Buck or Rainville references given in A046521, and my Aug 10 2017 comment there.
Recurrence for the row polynomials P_0(x) = 1, and P_n(x) = x*substitute(z=d/dx, exp(-z)/(1+z)) P_(n-1)(x), for n >= 1, with coefficient z^k of exp(-z)/(1+z) given by (-1)^k*A061354(k)/A061355(k). See the Roman reference Corollary 3.7.2., p. 50. (End)
The column sequences for the unsigned triangle Abs(T(n, k)), for k >= 2, are also given by {n^(n-k)*(n-1)*s(k-2, n)/(k-1)!}A049444.%20-%20_Wolfdieter%20Lang">{n>=k} with the row polynomials s(n, x) = risingfactorial(x - (n+1), n) of A049444. - _Wolfdieter Lang, Nov 21 2022

Better name by Peter Bala, Apr 08 2013

Edited by Joerg Arndt, Apr 08 2013

A245910 Number A(n,k) of pairs of endofunctions f, g on [n] satisfying f(g^k(i)) = f(i) for all i in [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 16, 1, 1, 10, 729, 1, 1, 12, 159, 65536, 1, 1, 10, 249, 3496, 9765625, 1, 1, 12, 207, 7744, 98345, 2176782336, 1, 1, 10, 249, 6856, 326745, 3373056, 678223072849, 1, 1, 12, 159, 9184, 302345, 17773056, 136535455, 281474976710656

Offset: 0

Alois P. Heinz, Aug 06 2014

			Square array A(n,k) begins:
0 :        1,     1,      1,      1,      1,      1, ...
1 :        1,     1,      1,      1,      1,      1, ...
2 :       16,    10,     12,     10,     12,     10, ...
3 :      729,   159,    249,    207,    249,    159, ...
4 :    65536,  3496,   7744,   6856,   9184,   3496, ...
5 :  9765625, 98345, 326745, 302345, 488745, 173225, ...

Alois P. Heinz, Antidiagonals n = 0..100, flattened

Columns k=0-10 give: A062206, A239761, A239777, A245912, A245913, A245914, A245915, A245916, A245917, A245918, A245919.
Main diagonal gives A245911.
Cf. A061356, A245980.

with(combinat):
b:= proc(n, i, k) option remember; unapply(`if`(n=0 or i=1, x^n,
      expand(add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
      x^(igcd(i, k)*j)*b(n-i*j, i-1, k)(x), j=0..n/i))), x)
    end:
A:= (n, k)-> `if`(k=0, n^(2*n), add(binomial(n-1, j-1)*n^(n-j)*
              b(j$2, k)(n), j=0..n)):
seq(seq(A(n, d-n), n=0..d), d=0..10);

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_, k_] := b[n, i, k] = Function[{x}, If[n == 0 || i == 1, x^n, Expand[Sum[(i-1)!^j*multinomial[n, Join[{ n-i*j}, Array[i&, j]]]/j!*x^(GCD[i, k]*j)*b[n-i*j, i-1, k][x], {j, 0, n/i}]]]]; A[0, ] = 1; A[n, k_] := If[k == 0, n^(2n), Sum[Binomial[n-1, j-1]*n^(n-j)* b[j, j, k][n], {j, 0, n}]]; Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 04 2015, after Alois P. Heinz *)

A007334 Number of spanning trees in the graph K_{n}/e, which results from contracting an edge e in the complete graph K_{n} on n vertices (for n>=2).

Original entry on oeis.org

1, 2, 8, 50, 432, 4802, 65536, 1062882, 20000000, 428717762, 10319560704, 275716983698, 8099130339328, 259492675781250, 9007199254740992, 336755653118801858, 13493281232954916864, 576882827135242335362, 26214400000000000000000

Offset: 2

N. J. A. Sloane

nonn

The old name (referring to the Chen-Goyal article) was "[Number of] essential complementary partitions of [an] n-set."
This sequence was obtained using the deletion-contraction recursions satisfied by the number of spanning trees for graphs. It is readily seen that the number of spanning trees in K_{n}-e (the complete graph K_{n} with an edge e deleted) is (n-2)*(n^{n-3}). Since the number of spanning trees in K_{n} is n^{n-2}, we see that (n-2)*(n^{n-3})+f(n)=n^{n-2} by the deletion-contraction recursion. Hence it follows that f(n)=2*n^{n-3}. - N. Eaton, W. Kook and L. Thoma (andrewk(AT)math.uri.edu), Jan 17 2004
With offset 0, the number of acyclic functions from {1,...,n} to {1,...,n+2}. See link below. - Dennis P. Walsh, Nov 27 2011
With offset 0, a(n) is the number of forests of rooted labeled trees on n nodes in which some (possibly all or none) of the trees have been specially designated. a(n) = Sum_{k=1..n} A061356(n,k)*2^k. E.g.f. is exp(T(x))^2 where T(x) is the e.g.f for A000169. The expected number of trees in each forest approaches 3 as n gets large. Cf. A225497. - Geoffrey Critzer, May 10 2013

			a(3)=2 because K_{3}/e consists of two vertices and two parallel edges, where each edge is a spanning tree.

J. Oxley, Matroid Theory, Oxford University Press, 1992.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alin Bostan, Frédéric Chyzak, Bérénice Delcroix-Oger, Guillaume Laplante-Anfossi, Vincent Pilaud, and Kurt Stoeckl, Diagonals of permutahedra and associahedra, Sém. Lotharingien Comb., 37th Formal Power Series Alg. Comb. (FPSAC 2025). See p. 7.
W.-K. Chen and I. C. Goyal, Tables of essential complementary partitions, IEEE Trans. Circuit Theory, 18 (1971), 562-563.
W.-K. Chen and I. C. Goyal, Tables of essential complementary partitions, IEEE Trans. Circuit Theory, 18 (1971), 562-563. (Annotated scanned copy)
Spencer Daugherty, Pamela E. Harris, Ian Klein, and Matt McClinton, Metered Parking Functions, arXiv:2406.12941 [math.CO], 2024. See pp. 19, 22.
N. Eaton, W. Kook and L. Thoma, Monotonicity for complete graphs, preprint, 2003.
Jean-Baptiste Priez and Aladin Virmaux, Non-commutative Frobenius characteristic of generalized parking functions: Application to enumeration, arXiv:1411.4161 [math.CO], 2014-2015.
Dennis Walsh, Notes on acyclic functions

The sequence is A058127(n, n-2) for n >= 2. - Peter Luschny, Apr 22 2009

Cf. A007830.

nn = 17; tx = Sum[n^(n - 1) x^n/n!, {n, 1, nn}];
Range[0, nn]! CoefficientList[Series[Exp[ tx]^2, {x, 0, nn}], x]  (* Geoffrey Critzer, May 10 2013 *)

{a(n)=if(n==2, 1, 1-polcoeff(sum(k=2, n-1, a(k)*x^k/(1+(k-1)*x+x*O(x^n))^(k-1)), n))} /* Paul D. Hanna, Jan 17 2013 */

a(n) = 2*n^{n-3} (n>=2).
E.g.f.: (-W(-x)/x)*exp(-W(-x)). - Paul Barry, Nov 19 2010 [With offset 0, and W = LambertW. Equals (W(-x)/(-x))^2 = (exp(-W(-x)))^2 (see a comment above). - Wolfdieter Lang, Nov 11 2022]
G.f.: Sum_{n>=1} a(n+1) * x^n / (1 + n*x)^n  =  x/(1-x). - Paul D. Hanna, Jan 17 2013

a(6) corrected and more terms from Sean A. Irvine, Dec 19 2017

After correction, this became identical (except for the offset) with A089104, contributed by N. Eaton, W. Kook and L. Thoma (andrewk(AT)math.uri.edu), Jan 17 2004. The two entries have been merged using the older A-number. - N. J. A. Sloane, Dec 19 2017

A368924 Triangle read by rows where T(n,k) is the number of labeled loop-graphs on n vertices with k loops and n-k non-loops such that it is possible to choose a different vertex from each edge.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 1, 9, 6, 1, 15, 68, 48, 12, 1, 222, 720, 510, 150, 20, 1, 3670, 9738, 6825, 2180, 360, 30, 1, 68820, 159628, 110334, 36960, 6895, 735, 42, 1, 1456875, 3067320, 2090760, 721560, 145530, 17976, 1344, 56, 1, 34506640, 67512798, 45422928, 15989232, 3402756, 463680, 40908, 2268, 72, 1

Offset: 0

Gus Wiseman, Jan 10 2024

The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

			Triangle begins:
      1
      0      1
      0      2      1
      1      9      6      1
     15     68     48     12      1
    222    720    510    150     20      1
   3670   9738   6825   2180    360     30      1
  68820 159628 110334  36960   6895    735     42      1
Row n = 3 counts the following loop-graphs:
  {{1,2},{1,3},{2,3}}  {{1},{1,2},{1,3}}  {{1},{2},{1,3}}  {{1},{2},{3}}
                       {{1},{1,2},{2,3}}  {{1},{2},{2,3}}
                       {{1},{1,3},{2,3}}  {{1},{3},{1,2}}
                       {{2},{1,2},{1,3}}  {{1},{3},{2,3}}
                       {{2},{1,2},{2,3}}  {{2},{3},{1,2}}
                       {{2},{1,3},{2,3}}  {{2},{3},{1,3}}
                       {{3},{1,2},{1,3}}
                       {{3},{1,2},{2,3}}
                       {{3},{1,3},{2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
Wikipedia, Axiom of choice.

Column k = n-1 is A002378.
The case of a unique choice is A061356, row sums A000272.
Column k = 0 is A137916, unlabeled version A137917.
Row sums appear to be A333331.
The complement has row sums A368596, covering case A368730.
The unlabeled version is A368926.
Without the choice condition we have A368928, A116508, A367863, A368597.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A014068 counts loop-graphs, unlabeled A000666.
Cf. A000169, A057500, A062740, A129271, A133686, A322661, A367869, A367902, A368601, A368835, A368836, A368927.

Table[Length[Select[Subsets[Subsets[Range[n],{1,2}],{n}], Count[#,{_}]==k&&Length[Select[Tuples[#], UnsameQ@@#&]]!=0&]],{n,0,5},{k,0,n}]

T(n)={my(t=-lambertw(-x + O(x*x^n))); [Vecrev(p) | p <- Vec(serlaplace(exp(-log(1-t)/2 - t/2 + t*y - t^2/4)))]}
{ my(A=T(8)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Jan 14 2024

E.g.f.: A(x,y) = exp(-log(1-T(x))/2 - T(x)/2 + y*T(x) - T(x)^2/4) where T(x) = -LambertW(-x) is the e.g.f. of A000169. - Andrew Howroyd, Jan 14 2024

a(36) onwards from Andrew Howroyd, Jan 14 2024

A074059 Dimension of the cohomology ring of the moduli space of n-pointed curves of genus 0 satisfying the associativity equations of physics (also known as the WDVV equations).

Original entry on oeis.org

1, 1, 2, 7, 34, 213, 1630, 14747, 153946, 1821473, 24087590, 352080111, 5636451794, 98081813581, 1843315388078, 37209072076483, 802906142007946, 18443166021077145, 449326835001457846, 11572432709175470807, 314160322966817351938, 8965995574654847062469

Offset: 1

Margaret A. Readdy, Aug 16 2002

nonn

			From _Paul D. Hanna_, Sep 24 2010: (Start)
E.g.f.: x + x^2/2! + 2*x^3/3! + 7*x^4/4! + 34*x^5/5! + 213*x^6/6! +...
The series reversion of the e.g.f. begins:
x - x^2/2 + x^3/6 - x^4/12 + x^5/20 - x^6/30 + x^7/42 - x^8/56 +... (End)

Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
V. M. Buchstaber and A. P. Veselov, Differential algebra of polytopes and inversion formulas, arXiv:2402.07168 [math.CO], 2024. See p. 9.
Brian Drake, Ira M. Gessel, and Guoce Xin, Three Proofs and a Generalization of the Goulden-Litsyn-Shevelev Conjecture on a Sequence Arising in Algebraic Geometry, J. of Integer Sequences, Vol. 10 (2007), #07.3.7.
I. P. Goulden, S. Litsyn, and V. Shevelev, On a Sequence Arising in Algebraic Geometry, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.7.
S. Keel, Intersection theory of moduli space of stable n-pointed curves of genus zero, Trans. Amer. Math. Soc. 330 (1992), 545-574.
M. Kontsevich and Y. Manin, Quantum cohomology of a product, (with Appendix by R. Kaufmann), Inv. Math. 124, f. 1-3 (1996) 313-339.
Curtis T. McMullen, Moduli spaces in genus zero and inversion of power series, (2012).
Margaret Readdy, The pre-WDVV ring of physics and its topology, arXiv:math/0511420 [math.CO], The Ramanujan Journal, Special issue on the Number Theory and Combinatorics in Physics, 10 (2005), 269-281.
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See pp. 12-13.

Row sums of triangle A074060.

series(exp(LambertW(-exp(-2)*(2+x))+2)-1,x,30): A:=simplify(%,symbolic): A074059:=n->n!*coeff(A,x,n): # Gessel

max = 19; $Assumptions = x > 0; (Series[ Exp[2 + ProductLog[-1, -(x+2)/E^2]] - 1, {x, 0, 19}] // CoefficientList[#, x] &) * Range[0, 19]! // Rest (* Jean-François Alcover, Jun 20 2013 *)

{a(n)=if(n<1,0,n!*polcoeff(serreverse(x-sum(k=2,n,(-x)^k/(k*(k-1)))+x*O(x^n)),n))} \\ Paul D. Hanna, Sep 24 2010

The exponential generating function A = A(x) = sum_{n>=1} a(n) x^n/n! satisfies the equation (1+A)log(1+A) = 2A-x. Explicitly, 1+A(x) = exp(2+W(e^(-2)(2+x))), where W is Lambert's W-function. - Ira M. Gessel, Dec 15 2005
E.g.f.: Series_Reversion[ x - Sum_{n>=2} (-x)^n/(n(n-1)) ]. - Paul D. Hanna, Sep 24 2010
Let h(x) = 1/(1-log(1+x)), then a(n) = ((h(x)*d/dx)^n)x evaluated at x=0, i.e., A(x) = exp(x*a(.)) = exp(x*h(u)*d/du) u, evaluated at u=0. Also, dA(x)/dx = h(A(x)). - Tom Copeland, Sep 06 2011
An o.g.f. is provided by the integral from w=0 to infinity of exp(-2w) * (1+z*w)^((1+z*w)/z). - Tom Copeland, Sep 09 2011
E.g.f. = -1/{1+W[-(2+x) exp(-2)]} with W(x) the Monir branch of the Lambert W fct. defined in A135338 and offset 0. - Tom Copeland, Oct 05 2011
a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator 1/(1-x)*exp(-x)*d/dx. Cf. A061356. - Peter Bala, Dec 08 2011
a(n) ~ n^(n-1) / (exp(1)*(exp(1)-2))^(n-1/2). - Vaclav Kotesovec, Oct 05 2013
a(1) = 1; a(n) = a(n-1) + Sum_{k=2..n-1} binomial(n-1,k) * a(k) * a(n-k). - Ilya Gutkovskiy, Aug 28 2020

More terms from Ira M. Gessel, Dec 15 2005

a(20)-a(22) from Stefano Spezia, Feb 14 2024

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

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jan 25 2001

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

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

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

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

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

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

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

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

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

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

A241980 Number of endofunctions on [n] where all cycle lengths are equal.

Original entry on oeis.org

1, 1, 4, 24, 206, 2300, 31742, 522466, 9996478, 218088504, 5344652492, 145386399554, 4347272984936, 141737636485588, 5004538251283846, 190247639729155110, 7747479351505166738, 336492490519027631984, 15526758954835131888980, 758548951300064645742034

Offset: 0

Alois P. Heinz, Aug 10 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..300

Cf. A005225, A061356, A212789, A242027 (column k=1).

Row sums of A243098.

with(numtheory):
b:= n-> `if`(n=0, 1, n!*add((d!*(n/d)^d)^(-1), d=divisors(n))):
a:= n-> add(binomial(n-1, j-1)*n^(n-j)*b(j), j=0..n):
seq(a(n), n=0..25);

nn=20;t[x_]:=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[Series[1+Sum[Exp[t[x]^i/i]-1,{i,1,nn}],{x,0,nn}],x] (* Geoffrey Critzer, Aug 11 2014 *)

a(n) = Sum_{j=0..n} C(n-1,j-1) * n^(n-j) * A005225(j).

a(n) = Sum_{k=0..n} A243098(n,k).

cp's OEIS Frontend

A139526 Triangle A061356 read right to left.

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A028421 Triangle read by rows: T(n, k) = (k+1)*A132393(n+1, k+1), for 0 <= k <= n.

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

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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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A245910 Number A(n,k) of pairs of endofunctions f, g on [n] satisfying f(g^k(i)) = f(i) for all i in [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A007334 Number of spanning trees in the graph K_{n}/e, which results from contracting an edge e in the complete graph K_{n} on n vertices (for n>=2).

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