A139526 -id:A139526 - cp's OEIS frontend

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

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

A251592 Triangle of coefficients of polynomials P(n,t) related to the Mittag-Leffler function, where P(n,t) = Product_{k=0..n-2} n*t-k.

Original entry on oeis.org

1, 0, 2, 0, -3, 9, 0, 8, -48, 64, 0, -30, 275, -750, 625, 0, 144, -1800, 7560, -12960, 7776, 0, -840, 13426, -77175, 204085, -252105, 117649, 0, 5760, -112896, 831488, -3010560, 5734400, -5505024, 2097152, 0, -45360, 1058508, -9573228

Offset: 1

Jean-François Alcover, Dec 05 2014

Second column (unsigned) 2, 3, 8, 30, 144, ... is A001048.

Diagonal 1, 2, 9, 64, 625, 7776, ... is A000169.

			Triangle begins :
  1;
  0,   2;
  0,  -3,     9;
  0,   8,   -48,   64;
  0, -30,   275, -750,    625;
  0, 144, -1800, 7560, -12960, 7776;
  ...

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 2nd ed. 1998

Peter Bala, Fractional iteration of a series inversion operator
MathOverflow, Properties and name of some polynomials
Eric Weisstein's MathWorld, Mittag-Leffler Function
Eric Weisstein's MathWorld, Wright Function

Cf. A000169, A001048, A156136, A000108 (B_2(x)), A001764 (B_3(x)), A002293 (B_4(x)), A002294 (B_5(x)), A002295 (B_6(x)), A002296 (B_7(x)), A007556 (B_8(x)), A062994 (B_9(x)), A059968 (B_10(x)), A230388 (B_11(x)), A139526, A260687.

P[n_, t_] := Product[n*t - k, {k, 0, n-2}]; row[n_] := CoefficientList[P[n, t], t]; Table[row[n], {n, 1, 10}] // Flatten

P(n,t) = (n-1)!*binomial(n*t, n-1).
From Peter Bala, Nov 15 2015: (Start)
E.g.f. (with constant term 1): B_t(x) = Sum_{n >= 0} 1/(n*t + 1)*binomial(n*t + 1,n)*x^n = 1 + x + 2*t*x^2/2! + 3*t(3*t - 1)*x^3/3! + 4*t*(4*t - 1)*(4*t - 2)*x^4/4! + ... is the generalized binomial series of Lambert. See Graham et al., Section 5.4 and Section 7.5.
In the notation of the Bala link, B_t(x) = I^t(1 + x) where I^t is a fractional inversion operator. B_(1+t)(x) is the e.g.f. for A260687.
B_t(x) = 1 + x*B_t(x)^t.
For complex r, B_t(x)^r = Sum_{n >= 0} r/(n*t + r)*binomial(n*t + r,n)*x^n.
log (B_t(x)) = Sum_{n >= 1} 1/(n*t)*binomial(n*t,n)*x^n.
B_2(x) is the o.g.f. for the Catalan numbers A000108. B_t(x) for t = 3,4,5,... gives the o.g.f. for various Fuss-Catalan sequences. See the cross references. (End)

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

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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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A251592 Triangle of coefficients of polynomials P(n,t) related to the Mittag-Leffler function, where P(n,t) = Product_{k=0..n-2} n*t-k.

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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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