A129062 -id:A129062 - cp's OEIS frontend

A000587 Rao Uppuluri-Carpenter numbers (or complementary Bell numbers): e.g.f. = exp(1 - exp(x)).

1, -1, 0, 1, 1, -2, -9, -9, 50, 267, 413, -2180, -17731, -50533, 110176, 1966797, 9938669, 8638718, -278475061, -2540956509, -9816860358, 27172288399, 725503033401, 5592543175252, 15823587507881, -168392610536153, -2848115497132448, -20819319685262839

Offset: 0

N. J. A. Sloane

Alternating row sums of Stirling2 triangle A048993.
Related to the matrix-exponential of the Pascal-matrix, see A000110 and A011971. - Gottfried Helms, Apr 08 2007
Closely linked to A000110 and especially the contribution there of Jonathan R. Love (japanada11(AT)yahoo.ca), Feb 22 2007, by offering what is a complementary finding.
Number of set partitions of 1..n with an even number of parts, minus the number of such partitions with an odd number of parts. - Franklin T. Adams-Watters, May 04 2010
After -2, the smallest prime is a(36) = -1454252568471818731501051, no others through a(100).  What is the first prime >0 in the sequence? - Jonathan Vos Post, Feb 02 2011
a(723) ~ 1.9*10^1265 is almost certainly prime. - D. S. McNeil, Feb 02 2011
Stirling transform of a(n) = [1, -1, 0, 1, 1, ...] is A033999(n) = [1, -1, 1, -1, 1, ...]. - Michael Somos, Mar 28 2012
Negated coefficients in the asymptotic expansion: A005165(n)/n! ~ 1 - 1/n + 1/n^2 + 0/n^3 - 1/n^4 - 1/n^5 + 2/n^6 + 9/n^7 + 9/n^8 - 50/n^9 - 267/n^10 - 413/n^11 + O(1/n^12), starting from the O(1/n) term. - Vladimir Reshetnikov, Nov 09 2016
Named after Venkata Ramamohana Rao Uppuluri and John A. Carpenter of the Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee. They are called "Rényi numbers" by Fekete (1999), after the Hungarian mathematician Alfréd Rényi (1921-1970). - Amiram Eldar, Mar 11 2022

			G.f. = 1 - x + x^3 + x^4 - 2*x^5 - 9*x^6 - 9*x^7 + 50*x^8 + 267*x^9 + 413*x^10 - ...

N. A. Kolokolnikova, Relations between sums of certain special numbers (Russian), in Asymptotic and enumeration problems of combinatorial analysis, pp. 117-124, Krasnojarsk. Gos. Univ., Krasnoyarsk, 1976.
Alfréd Rényi, Új modszerek es eredmenyek a kombinatorikus analfzisben. I. MTA III Oszt. Ivozl., Vol. 16 (1966), pp. 7-105.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
M. V. Subbarao and A. Verma, Some remarks on a product expansion. An unexplored partition function, in Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics (Gainesville, FL, 1999), pp. 267-283, Kluwer, Dordrecht, 2001.

Alois P. Heinz, Table of n, a(n) for n = 0..595 (first 101 terms from T. D. Noe)
M. Aguiar and A. Lauve, The characteristic polynomial of the Adams operators on graded connected Hopf algebras, 2014. See Example 31. - _N. J. A. Sloane_, May 24 2014
W. Asakly, A. Blecher, C. Brennan, A. Knopfmacher, T. Mansour, and S. Wagner, Set partition asymptotics and a conjecture of Gould and Quaintance, Journal of Mathematical Analysis and Applications, Volume 416, Issue 2 (15 August 2014), Pages 672-682.
Tewodros Amdeberhan, Valerio de Angelis and Victor H. Moll, Complementary Bell numbers: arithmetical properties and Wilf's conjecture.
S. Barbero, U. Cerruti, and N. Murru, A Generalization of the Binomial Interpolated Operator and its Action on Linear Recurrent Sequences, J. Int. Seq., Vol. 13 (2010), Article 10.9.7.
R. E. Beard, On the Coefficients in the Expansion of e^(e^t) and e^(-e^t), J. Institute of Actuaries, Vol. 76 (1950), pp. 152-163. [Annotated scanned copy]
Pascal Caron, Jean-Gabriel Luque, Ludovic Mignot, and Bruno Patrou, State complexity of catenation combined with a boolean operation: a unified approach, arXiv:1505.03474 [cs.FL], 2015.
Valerio De Angelis and Dominic Marcello, Wilf's Conjecture, The American Mathematical Monthly, Vol. 123, No. 6 (2016), pp. 557-573.
S. de Wannemacker, T. Laffey and R. Osburn, On a conjecture of Wilf, arXiv:math/0608085 [math.NT], 2006-2007.
Branko Dragovich, On Summation of p-Adic Series, arXiv:1702.02569 [math.NT], 2017.
Branko Dragovich, Andrei Yu. Khrennikov, and Natasa Z. Misic, Summation of p-Adic Functional Series in Integer Points, arXiv:1508.05079, 2015
B. Dragovich and N. Z. Misic, p-Adic invariant summation of some p-adic functional series, P-Adic Numbers, Ultrametric Analysis, and Applications, Volume 6, Issue 4 (October 2014), pp. 275-283.
Antal E. Fekete, Apropos Bell and Stirling Numbers, Crux Mathematicorum, Vol. 25, No. 5 (1999), pp. 274-281.
B. Harris and L. Schoenfeld, Asymptotic expansions for the coefficients of analytic functions, Ill. J. Math., Vol. 12 (1968), pp. 264-277.
M. Klazar, Counting even and odd partitions, Amer. Math. Monthly, Vol. 110, No. 6 (2003), pp. 527-532.
M. Klazar, Bell numbers, their relatives and algebraic differential equations, J. Combin. Theory, A 102 (2003), 63-87.
A. Knopfmacher and M. E. Mays, Graph compositions I: Basic enumerations, Integers, Vol. 1 (2001), Article A4. (See the first two columns of the table on p. 9.)
Vaclav Kotesovec, Plot of |a(n)/n!|^(1/n) / |exp(1/W(-n))/W(-n)| for n = 1..40000, where W is the LambertW function.
Peter J. Larcombe, Jack Sutton, and James Stanton, A note on the constant 1/e, Palest. J. Math. (2023) Vol. 12, No. 2, 609-619. See p. 617.
J. W. Layman and C. L. Prather, Generalized Bell numbers and zeros of successive derivatives of an entire function, Journal of Mathematical Analysis and Applications, Volume 96, Issue 1 (15 October 1983), Pages 42-51.
Toufik Mansour and Mark Shattuck, Counting subword patterns in permutations arising as flattened partitions of sets, Appl. Anal. Disc. Math. (2022), OnLine-First (00):9-9.
T. Mansour, M. Shattuck and D. G. L. Wang, Recurrence relations for patterns of type (2, 1) in flattened permutations, arXiv preprint arXiv:1306.3355 [math.CO], 2013.
S. Ramanujan, Notebook entry.
V. R. Rao Uppuluri and J. A. Carpenter, Numbers generated by the function exp(1-e^x), Fib. Quart., Vol. 7, No. 4 (1969), pp. 437-448.
Frank Ruskey and Jennifer Woodcock, The Rand and block distances of pairs of set partitions, Combinatorial algorithms, 287-299, Lecture Notes in Comput. Sci., 7056, Springer, Heidelberg, 2011.
Frank Ruskey, Jennifer Woodcock and Yuji Yamauchi, Counting and computing the Rand and block distances of pairs of set partitions, Journal of Discrete Algorithms, Volume 16 (October 2012), Pages 236-248. [_N. J. A. Sloane_, Oct 03 2012]
M. Z. Spivey, On Solutions to a General Combinatorial Recurrence, J. Int. Seq., Vol. 14 (2011), Article 11.9.7.
D. Subedi, Complementary Bell Numbers and p-adic Series, J. Int. Seq., Vol. 17 (2014), Article 14.3.1.
A. Vieru, Agoh's conjecture: its proof, its generalizations, its analogues, arXiv preprint arXiv:1107.2938 [math.NT], 2011.
Eric Weisstein's World of Mathematics, Complementary Bell Number.
D. Wuilquin, Letters to N. J. A. Sloane, August 1984.
Yifan Yang, On a multiplicative partition function, Electron. J. Combin., Vol. 8, No. 1 (2001), Research Paper 19.

Cf. A000110, A011971 (base triangle PE), A078937 (PE^2).

Cf. A007318, A153229, A213170.

a000587 n = a000587_list !! n
a000587_list = 1 : f a007318_tabl [1] where
   f (bs:bss) xs = y : f bss (y : xs) where y = - sum (zipWith (*) xs bs)
-- Reinhard Zumkeller, Mar 04 2014

b:= proc(n, t) option remember; `if`(n=0, 1-2*t,
      add(b(n-j, 1-t)*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..35);  # Alois P. Heinz, Jun 28 2016

Table[ -1 * Sum[ (-1)^( k + 1) StirlingS2[ n, k ], {k, 0, n} ], {n, 0, 40} ]
With[{nn=30},CoefficientList[Series[Exp[1-Exp[x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 04 2011 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ 1 - Exp[x]], {x, 0, n}]]; (* Michael Somos, May 27 2014 *)
a[ n_] := If[ n < 0, 0, With[{m = n + 1}, SeriesCoefficient[ Series[ Nest[ x Factor[ 1 - # /. x -> x / (1 - x)] &, 0, m], {x, 0, m}], {x, 0, m}]]]; (* Michael Somos, May 27 2014 *)
Table[BellB[n, -1], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 20 2015 *)
b[1] = 1; k = 1; Flatten[{1, Table[Do[j = k; k -= b[m]; b[m] = j;, {m, 1, n-1}]; b[n] = k; k*(-1)^n, {n, 1, 40}]}] (* Vaclav Kotesovec, Sep 09 2019 *)

{a(n) = if( n<0, 0, n! * polcoeff( exp( 1 - exp( x + x * O(x^n))), n))}; /* Michael Somos, Mar 14 2011 */

{a(n) = local(A); if( n<0, 0, n++; A = O(x); for( k=1, n, A = x - x * subst(A, x, x / (1 - x))); polcoeff( A, n))}; /* Michael Somos, Mar 14 2011 */

Vec(serlaplace(exp(1 - exp(x+O(x^99))))) /* Joerg Arndt, Apr 01 2011 */

a(n)=round(exp(1)*suminf(k=0,(-1)^k*k^n/k!))
vector(20,n,a(n-1)) \\ Derek Orr, Sep 19 2014 -- a direct approach

x='x+O('x^66); Vec(serlaplace(exp(1 - exp(x)))) \\ Michel Marcus, Sep 19 2014

# The objective of this implementation is efficiency.
# n -> [a(0), a(1), ..., a(n)] for n > 0.
def A000587_list(n):
    A = [0 for i in range(n)]
    A[n-1] = 1
    R = [1]
    for j in range(0, n):
        A[n-1-j] = -A[n-1]
        for k in range(n-j, n):
            A[k] += A[k-1]
        R.append(A[n-1])
    return R
# Peter Luschny, Apr 18 2011

# Python 3.2 or higher required
from itertools import accumulate
A000587, blist, b = [1,-1], [1], -1
for _ in range(30):
    blist = list(accumulate([b]+blist))
    b = -blist[-1]
    A000587.append(b) # Chai Wah Wu, Sep 19 2014

expnums(26, -1) # Zerinvary Lajos, May 15 2009

a(n) = e*Sum_{k>=0} (-1)^k*k^n/k!. - Benoit Cloitre, Jan 28 2003
E.g.f.: exp(1 - e^x).
a(n) = Sum_{k=0..n} (-1)^k S2(n, k), where S2(i, j) are the Stirling numbers of second kind A008277.
G.f.: (x/(1-x))*A(x/(1-x)) = 1 - A(x); the binomial transform equals the negative of the sequence shifted one place left. - Paul D. Hanna, Dec 08 2003
With different signs: g.f.: Sum_{k>=0} x^k/Product_{L=1..k} (1 + L*x).
Recurrence: a(n) = -Sum_{i=0..n-1} a(i)*C(n-1, i). - Ralf Stephan, Feb 24 2005
Let P be the lower-triangular Pascal-matrix, PE = exp(P-I) a matrix-exponential in exact integer arithmetic (or PE = lim exp(P)/exp(1) as limit of the exponential); then a(n) = PE^-1 [n,1]. - Gottfried Helms, Apr 08 2007
Take the series 0^n/0! - 1^n/1! + 2^n/2! - 3^n/3! + 4^n/4! + ... If n=0 then the result will be 1/e, where e = 2.718281828... If n=1, the result will be -1/e. If n=2, the result will be 0 (i.e., 0/e). As we continue for higher natural number values of n sequence for the Roa Uppuluri-Carpenter numbers is generated in the numerator, i.e., 1/e, -1/e, 0/e, 1/e, 1/e, -2/e, -9/e, -9/e, 50/e, 267/e, ... . - Peter Collins (pcolins(AT)eircom.net), Jun 04 2007
The sequence (-1)^n*a(n), with general term Sum_{k=0..n} (-1)^(n-k)*S2(n, k), has e.g.f. exp(1-exp(-x)). It also has Hankel transform (-1)^C(n+1,2)*A000178(n) and binomial transform A109747. - Paul Barry, Mar 31 2008
G.f.: 1 / (1 + x / (1 - x / (1 + x / (1 - 2*x / (1 + x / (1 - 3*x / (1 + x / ...))))))). - Michael Somos, May 12 2012
From Sergei N. Gladkovskii, Sep 28 2012 to Feb 07 2014: (Start)
Continued fractions:
G.f.: -1/U(0) where U(k) = x*k - 1 - x + x^2*(k+1)/U(k+1).
G.f.: 1/(U(0)+x) where U(k) = 1 + x - x*(k+1)/(1 + x/U(k+1)).
G.f.: 1+x/G(0) where G(k) = x*k - 1 + x^2*(k+1)/G(k+1).
G.f.: (1 - G(0))/(x+1) where G(k) = 1 - 1/(1-k*x)/(1-x/(x+1/G(k+1) )).
G.f.: 1 + x/(G(0)-x) where G(k) = x*k + 2*x - 1 - x*(x*k+x-1)/G(k+1).
G.f.: G(0)/(1+x), where G(k) = 1-x^2*(k+1)/(x^2*(k+1)+(x*k-1-x)*(x*k-1)/G(k+1)).
(End)
a(n) = B_n(-1), where B_n(x) is n-th Bell polynomial. - Vladimir Reshetnikov, Oct 20 2015
From Mélika Tebni, May 20 2022: (Start)
a(n) = Sum_{k=0..n} (-1)^k*Bell(k)*A129062(n, k).
a(n) = Sum_{k=0..n} (-1)^k*k!*A130191(n, k). (End)

A209849 Triangle read by rows: coefficients of polynomials in Sum_{k = 0..t} k^n * binomial(t,k).

Original entry on oeis.org

1, 1, 1, 0, 3, 1, -2, 3, 6, 1, 0, -10, 15, 10, 1, 16, -30, -15, 45, 15, 1, 0, 112, -210, 35, 105, 21, 1, -272, 588, 28, -735, 280, 210, 28, 1, 0, -2448, 5292, -2436, -1575, 1008, 378, 36, 1, 7936, -18960, 4140, 20160, -14595, -1575, 2730, 630, 45, 1

Offset: 1

Peter Bala, Mar 15 2012

Repeatedly applying the operator x*d/dx to (1 + x)^t (t a nonnegative integer) and evaluating at x = 1 yields Sum_{k = 0..t} k^n*binomial(t,k) = R(n,t)*2^(t-n), where R(n,t) is a polynomial in t for n = 1,2,.... The polynomial sequence {R(n,t)}_{n>=0} is of binomial type. The first few values are given in the example section below.
This triangle lists the coefficients of these polynomials in ascending powers of t (omitting R(0,t) = 1). A closely related triangle is A102573, which lists the coefficients of the polynomials R(n,t) after factors of t and t*(1 + t) have been removed.
This is the case m = 2 of a family of binomial type polynomials satisfying the recurrence R(n+1,t) = t*(m*(R(n,t) - R(n,t-1)) + R(n,t-1)) with R(0,t) = 1. Case m = 0 gives the falling factorials (A008275); Case m = -1 gives a signed version of A079641.

			Repeatedly applying the operator x*d/dx to (1 + x)^n and evaluating the result at x = 1 yields
Sum_{k = 0..n} k   * binomial(n,k) =  n                * 2^(n-1).
Sum_{k = 0..n} k^2 * binomial(n,k) = (n +   n^2)       * 2^(n-2).
Sum_{k = 0..n} k^3 * binomial(n,k) = (    3*n^2 + n^3) * 2^(n-3).
Triangle begins:
  n\k|    1     2     3     4     5     6     7     8
  = = = = = = = = = = = = = = = = = = = = = = = = = =
  1  |    1
  2  |    1     1
  3  |    0     3     1
  4  |   -2     3     6     1
  5  |    0   -10    15    10     1
  6  |   16   -30   -15    45    15     1
  7  |    0   112  -210    35   105    21     1
  8  | -272   588    28  -735   280   210    28     1
  ...

Seiichi Manyama, Rows n = 1..140, flattened

Columns k=1..3 give A155585(n-1), A383165, A383166.

Cf. A008275, A009006 (alt. row sums), A059419, A063376, A075497, A079641, A102573, A119468, A129062, A154602, A176668, A195204, A383140.

# The function BellMatrix is defined in A264428.
g := n -> 2^n*euler(n,1): BellMatrix(g, 9); # Peter Luschny, Jan 21 2016

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[2^# EulerE[#, 1]&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)

T(n, k) = sum(j=k, n, 2^(n-j)*stirling(n, j, 2)*stirling(j, k, 1)); \\ Seiichi Manyama, Apr 16 2025

# uses[bell_matrix from A264428]
g = lambda n: sum((-2)^(n-k)*factorial(k)*stirling_number2(n,k) for k in (0..n))
bell_matrix(g, 9) # Peter Luschny, Jan 21 2016

def a_row(n):
    s = sum(2^(n-k)*stirling_number2(n, k)*falling_factorial(x, k) for k in (0..n))
    return expand(s).list()[1:]
for n in (1..10): print(a_row(n)) # Seiichi Manyama, Apr 16 2025

T(n,k) = Sum_{j = 0..n} (-1)^(n+k) * (-2)^(n-j) * Stirling2(n,j) * |Stirling1(j,k)|. [corrected by Seiichi Manyama, Apr 16 2025]
E.g.f.: F(x,t) := (1/2 + 1/2*exp(2*x))^t = (1 + tanh(-x))^(-t) = 1 + t*x + (t+t^2)*x^2/2! + (3*t^2+t^3)*x^3/3! + ... satisfies the delay differential equation d/dx(F(x,t)) = 2*F(x,t) - F(x,t-1).
Recurrence for row polynomials R(n,t): R(n+1,t) = t*(2*R(n,t) - R(n,t-1)) with R(0,t) = 1.
Let D be the backward difference operator D(f(x)) = f(x) - f(x-1). Then (x*D)^n(2^x) = 2^(x-n)*R(n,x). Cf. A079641.
Discrete Dobinski-type relation: R(n,x) = 1/2^x*Sum_{k = 0..inf} (2*k)^n*x*(x - 1)*...*(x - k + 1)/k!, valid for x = 0,1,2,.... and n >= 1.
Other Dobinski-type relations: exp(-x)*Sum_{k = 0..inf} R(n,k)*x^k/k! = n-th row polynomial of A075497.
exp(-x)*Sum_{k = 0..inf} R(n,k+1)*x^k/k! = n-th row polynomial of A154602.
i^(-n)*exp(i*x)*Sum_{k = 0..inf} R(n,-k)*(-i*x)^k/k! = n-th row polynomial of A059419 where i = sqrt(-1).
Writing x^[n] in place of R(n,x) we have the analog of the Bernoulli summation formula for powers of integers: Sum_{k = 1..n-1} k^[p] = 1/(p + 1)*Sum_{k = 0..p} 2^k*binomial(p+1,k)*B_k*n^[p+1-k], where B_k = [1,-1/2,1/6,0,-1/30,...] is the sequence of Bernoulli numbers.
n-th row sum R(n,1) equals 2^(n-1). Alternating row sums R(n,-1) starting [-1,0,2,0,-16,0,272,...] are signed tangent numbers - see A009006 and A155585.
R(n+1,2) = 2^n + 4^n = A063376(n).
Triangle as a product of lower triangular arrays equals A075497*A008275.
The triangle of connection constants between the polynomials (x + 1)^[n] and x^[n] appears to be A119468 = (P^2 + 1)/2, where P denotes Pascal's triangle.
Also the Bell transform of the sequence 2^n*E(n,1), E(n,x) the Euler polynomials (A155585). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 21 2016
From Peter Bala, Jun 26 2016: (Start)
With row and column numbering starting at 0:
E.g.f. is exp(x)/cosh(x)*((1 + exp(2*x))/2)^t = 1 + (1 + t)*x + (3*t + t^2)*x^2/2! + (-2 + 3*t + 6*t^2 + t^3)*x^3/3! + ....
Exponential Riordan array [d/dx(f(x)), f(x)] belonging to the Derivative subgroup of the Riordan group, where f(x) = log((1 + exp(2*x))/2) and df/dx = exp(x)/cosh(x) is the e.g.f. for A155585. (End)
T(n,k) = [x^k] Sum_{k=0..n} 2^(n-k) * Stirling2(n,k) * FallingFactorial(x,k). - Seiichi Manyama, Apr 16 2025
E.g.f. of column k (with leading zeros): f(x)^k / k! with f(x) = log(1 + (exp(2*x) - 1)/2). - Seiichi Manyama, Apr 18 2025

A079641 Matrix product of Stirling2-triangle A008277(n,k) and unsigned Stirling1-triangle |A008275(n,k)|.

Original entry on oeis.org

1, 2, 1, 6, 6, 1, 26, 36, 12, 1, 150, 250, 120, 20, 1, 1082, 2040, 1230, 300, 30, 1, 9366, 19334, 13650, 4270, 630, 42, 1, 94586, 209580, 166376, 62160, 11900, 1176, 56, 1, 1091670, 2562354, 2229444, 952728, 220500, 28476, 2016, 72, 1, 14174522

Offset: 1

Vladeta Jovovic, Jan 30 2003

Triangle T(n,k), 1<=k<=n, read by rows, given by (0, 2, 1, 4, 2, 6, 3, 8, 4, 10, 5, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 22 2011
Subtriangle of triangle in A129062. - Philippe Deléham, Feb 17 2013
Also the Bell transform of A000629. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

			Triangle begins:
  1;
  2,1;
  6,6,1;
  26,36,12,1;
  150,250,120,20,1;
  1082,2040,1230,300,30,1;
  ...
Triangle (0,2,1,4,2,6,3,8,4,...) DELTA (1,0,1,0,1,0,1,0,1,...) begins:
  1
  0, 1
  0, 2, 1
  0, 6, 6, 1
  0, 26, 36, 12, 1
  0, 150, 250, 120, 20, 1
  0, 1082, 2040, 1230, 300, 30, 1. - _Philippe Deléham_, Dec 22 2011

Nick Early, Canonical Bases for Permutohedral Plates, arXiv:1712.08520 [math.CO], 2017.
Nick Early, Honeycomb tessellations and canonical bases for permutohedral blades, arXiv:1810.03246 [math.CO], 2018.
D. E. Knuth, Convolution polynomials, arXiv:math/9207221 [math.CA], 1992; The Mathematica J., 2 (1992), 67-78.

Cf. A000670 (row sums), A000629 (first column), A195204, A195205. A209849, A129062

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> add((-1)^(n-k)*2^k*k!*combinat:-stirling2(n, k), k=0..n), 9); # Peter Luschny, Jan 26 2016

rows = 10;
t = Table[Sum[(-1)^(n-k)*2^k*k!*StirlingS2[n, k], {k,0,n}], {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)

T(n, k) = Sum_{i=k..n} A008277(n, i) * |A008275(i, k)|.
E.g.f.: (2-exp(x))^(-y). - Vladeta Jovovic, Nov 22 2003
From Peter Bala, Sep 12 2011: (Start)
The row generating polynomials R(n,x) begin R(1,x) = x, R(2,x) = 2*x + x^2, R(3,x) = 6*x + 6*x^2 + x^3 and satisfy the recurrence R(n+1,x) = x*(2*R(n,x+1) - R(n,x)). They form a sequence of binomial type polynomials. In particular, denoting R(n,x) by x^[n] to emphasize the analogies with the monomial polynomials x^n, we have the binomial expansion (x + y)^[n] = Sum_{k = 0..n} binomial(n,k)*x^[n-k]*y^[k].
There is a Dobinski-type formula: exp(-x)*Sum_{k >= 0} (-k)^[n] * x^k/k! = Bell(n,-x). The alternating n-th row entries (-1)^k * T(n,k) are the connection coefficients expressing the polynomial Bell(n,-x) as a linear combination of Bell(k,x), 1 <= k <= n. For example, the list of coefficients of R(4,x) is [26, 36, 12, 1] and we have Bell(4,-x) = -26*Bell(1,x) + 36*Bell(2,x) - 12*Bell(3,x) + Bell(4,x).
The row polynomials also satisfy an analog of the Bernoulli's summation formula for powers of integers, namely, Sum_{k = 1..n} k^[p] = 1/(p+1) * Sum_{k = 0..p} binomial(p+1,k) * B_k * n^[p+1-k], where B_k denotes the Bernoulli numbers. Compare with A195204 and A195205. (End)
Let D be the forward difference operator D(f(x)) = f(x+1) - f(x). Then the n-th row polynomial R(n,x) = 1/f(x) * (x*D)^n(f(x)) with f(x) = 2^x. Cf. A209849. Also cf. A008277, where the row polynomials are given by 1/f(x) * (x*d/dx)^n(f(x)), where now f(x) = exp(x). - Peter Bala, Mar 16 2012
Conjecture: o.g.f. as a continued fraction of Stieltjes type: 1/(1 - x*z/(1 - 2*z/(1 - (x + 1)*z/(1 - 4*z/(1 - (x + 2)*z/(1 - 6*z/(1 - (x + 3)*z/(1 - 8*z/(1 - ... ))))))))) =  1 + x*z + (2*x + x^2)*z^2 + (6*x + 6*x^2 + x^3)*z^3 + .... - Peter Bala, Dec 12 2024

A325872 T(n, k) = [x^k] Sum_{k=0..n} Stirling1(n, k)*FallingFactorial(x, k), triangle read by rows, for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, -2, 1, 0, 7, -6, 1, 0, -35, 40, -12, 1, 0, 228, -315, 130, -20, 1, 0, -1834, 2908, -1485, 320, -30, 1, 0, 17582, -30989, 18508, -5005, 665, -42, 1, 0, -195866, 375611, -253400, 81088, -13650, 1232, -56, 1, 0, 2487832, -5112570, 3805723, -1389612, 279048, -32130, 2100, -72, 1

Offset: 0

Peter Luschny, Jun 27 2019

			Triangle starts:
[0] [1]
[1] [0,       1]
[2] [0,      -2,        1]
[3] [0,       7,       -6,       1]
[4] [0,     -35,       40,     -12,        1]
[5] [0,     228,     -315,     130,      -20,      1]
[6] [0,   -1834,     2908,   -1485,      320,    -30,      1]
[7] [0,   17582,   -30989,   18508,    -5005,    665,    -42,    1]
[8] [0, -195866,   375611, -253400,    81088, -13650,   1232,  -56,   1]
[9] [0, 2487832, -5112570, 3805723, -1389612, 279048, -32130, 2100, -72, 1]

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened).
Gabriella Bretti, Pierpaolo Natalini and Paolo E. Ricci, A new set of Sheffer-Bell polynomials and logarithmic numbers, Georgian Mathematical Journal, Feb. 2019, page 8.
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.

Columns k=0..3 give A000007, (-1)^(n+1) * A003713(n), (-1)^n * A341587(n), (-1)^(n+1) * A341588(n).

Cf. A039814 (variant), A129062, A325873.

p[n_] := Sum[StirlingS1[n, k] FactorialPower[x, k] , {k, 0, n}];
Table[CoefficientList[FunctionExpand[p[n]], x], {n, 0, 9}] // Flatten

T(n, k) = sum(j=k, n, stirling(n, j, 1)*stirling(j, k, 1)); \\ Seiichi Manyama, Apr 18 2025

def a_row(n):
    s = sum((-1)^(n-k)*stirling_number1(n,k)*falling_factorial(x,k) for k in (0..n))
    return expand(s).list()
[a_row(n) for n in (0..9)]

From Seiichi Manyama, Apr 18 2025: (Start)
T(n,k) = Sum_{j=k..n} Stirling1(n,j) * Stirling1(j,k).
E.g.f. of column k (with leading zeros): f(x)^k / k! with f(x) = log(1 + log(1 + x)). (End)

A326477 Coefficients of polynomials related to ordered set partitions. Triangle read by rows, T_{m}(n, k) for m = 2 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 4, 3, 0, 46, 60, 15, 0, 1114, 1848, 840, 105, 0, 46246, 88770, 54180, 12600, 945, 0, 2933074, 6235548, 4574130, 1469160, 207900, 10395, 0, 263817646, 605964450, 505915410, 199849650, 39729690, 3783780, 135135

Offset: 0

Peter Luschny, Jul 08 2019

			Triangle starts:
[0] [1]
[1] [0, 1]
[2] [0, 4, 3]
[3] [0, 46, 60, 15]
[4] [0, 1114, 1848, 840, 105]
[5] [0, 46246, 88770, 54180, 12600, 945]
[6] [0, 2933074, 6235548, 4574130, 1469160, 207900, 10395]

Row sums A094088. Alternating row sums A153881 starting at 0.
Main diagonal A001147. Associated set partitions A241171.
A129062 (m=1, associated with A131689), this sequence (m=2), A326587 (m=3, associated with A278073), A326585 (m=4, associated with A278074).

CL := f -> PolynomialTools:-CoefficientList(f, x):
FL := s -> ListTools:-Flatten(s, 1):
StirPochConv := proc(m, n) local P, L; P := proc(m, n) option remember;
`if`(n = 0, 1, add(binomial(m*n, m*k)*P(m, n-k)*x, k=1..n)) end:
L := CL(P(m, n)); CL(expand(add(L[k+1]*pochhammer(x,k)/k!, k=0..n))) end:
FL([seq(StirPochConv(2,n), n = 0..7)]);

P[, 0] = 1; P[m, n_] := P[m, n] = Sum[Binomial[m*n, m*k]*P[m, n-k]*x, {k, 1, n}] // Expand;
T[m_][n_] := CoefficientList[P[m, n], x].Table[Pochhammer[x, k]/k!, {k, 0, n}] // CoefficientList[#, x]&;
Table[T[2][n], {n, 0, 7}] // Flatten (* Jean-François Alcover, Jul 21 2019 *)

def StirPochConv(m, n):
    z = var('z'); R = ZZ[x]
    F = [i/m for i in (1..m-1)]
    H = hypergeometric([], F, (z/m)^m)
    P = R(factorial(m*n)*taylor(exp(x*(H-1)), z, 0, m*n + 1).coefficient(z, m*n))
    L = P.list()
    S = sum(L[k]*rising_factorial(x,k) for k in (0..n))
    return expand(S).list()
for n in (0..6): print(StirPochConv(2, n))

For m >= 1 let P(m,0) = 1 and P(m, n) = Sum_{k=1..n} binomial(m*n, m*k)*P(m, n-k)*x for n > 0. Then T_{m}(n, k) = Sum_{k=0..n} ([x^k]P(m, n))*rf(x,k)/k! where rf(x,k) are the rising factorial powers. T(n, k) = T_{2}(n, k).

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A129063 Third column (m=2) of triangle A129062 and second column of triangle A079641.

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A129064 Fourth column (m=3) of triangle A129062 and third column of triangle A079641.

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A000587 Rao Uppuluri-Carpenter numbers (or complementary Bell numbers): e.g.f. = exp(1 - exp(x)).

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A209849 Triangle read by rows: coefficients of polynomials in Sum_{k = 0..t} k^n * binomial(t,k).

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A079641 Matrix product of Stirling2-triangle A008277(n,k) and unsigned Stirling1-triangle |A008275(n,k)|.

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A325872 T(n, k) = [x^k] Sum_{k=0..n} Stirling1(n, k)*FallingFactorial(x, k), triangle read by rows, for n >= 0 and 0 <= k <= n.

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A326477 Coefficients of polynomials related to ordered set partitions. Triangle read by rows, T_{m}(n, k) for m = 2 and 0 <= k <= n.

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