A092552 -id:A092552 - cp's OEIS frontend

A220181 E.g.f.: Sum_{n>=0} (1 - exp(-n*x))^n.

1, 1, 7, 115, 3451, 164731, 11467387, 1096832395, 138027417451, 22111390122811, 4393756903239067, 1060590528331645675, 305686632592587314251, 103695663062502304228891, 40895823706632785802087547, 18554695374154504939196298955, 9596336362873294022956267703851

Offset: 0

Paul D. Hanna, Dec 06 2012

Compare to the trivial identity: exp(x) = Sum_{n>=0} (1 - exp(-x))^n.
Compare to the e.g.f. of A092552: Sum_{n>=1} (1 - exp(-n*x))^n/n.
From Arvind Ayyer, Oct 25 2020: (Start)
a(n) is also the number of acyclic orientations with unique sink of the complete bipartite graph K_{n,n+1}
a(n) is also the number of toppleable permutations in S_{2n}. A toppleable permutation pi in S_{2n} satisfies pi_i <= n-1+i for 1 <= i <= n+1 and pi_i >= i-n for n+2 <= i <= 2n. (End)
Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p for n >= 1 is purely periodic with period p - 1. For example, modulo 7 the sequence becomes [1, 0, 3, 0, 0, 1, 1, 0, 3, 0, 0, 1, 1, 0, 3, 0, 0, 1 ...], with an apparent period of 6. Cf. A122399. - Peter Bala, Jun 01 2022

			O.g.f.: F(x) = 1 + x + 7*x^2 + 115*x^3 + 3451*x^4 + 164731*x^5 +...
where F(x) = 1 + x/(1+x) + 2^2*2!*x^2/((1+2*1*x)*(1+2*2*x)) + 3^3*3!*x^3/((1+3*1*x)*(1+3*2*x)*(1+3*3*x)) + 4^4*4!*x^4/((1+4*1*x)*(1+4*2*x)*(1+4*3*x)*(1+4*4*x)) +...
...
E.g.f.: A(x) = 1 + x + 7*x^2/2! + 115*x^3/3! + 3451*x^4/4! + 164731*x^5/5! +...
where the e.g.f. satisfies the identities:
(1) A(x) = 1 + (1-exp(-x)) + (1-exp(-2*x))^2 + (1-exp(-3*x))^3 + (1-exp(-4*x))^4 + (1-exp(-5*x))^5 + (1-exp(-6*x))^6 +...
(2) A(x) = exp(-x) + exp(-2*x)*(1-exp(-2*x)) + exp(-3*x)*(1-exp(-3*x))^2 + exp(-4*x)*(1-exp(-4*x))^3 + exp(-5*x)*(1-exp(-5*x))^4 + exp(-6*x)*(1-exp(-6*x))^5 +...
(3) 2*A(x) = 2 + (1-exp(-2*x)) + (1-exp(-3*x))^2 + (1-exp(-4*x))^3 + (1-exp(-5*x))^4 + (1-exp(-6*x))^5 + (1-exp(-7*x))^6 +...
E.g.f. at offset=1 begins:
B(x) = x + x^2/2! + 7*x^3/3! + 115*x^4/4! + 3451*x^5/5! + 164731*x^6/6! +...
where
B(x) = (1-exp(-x)) + (1-exp(-2*x))^2/2^2 + (1-exp(-3*x))^3/3^2 + (1-exp(-4*x))^4/4^2 + (1-exp(-5*x))^5/5^2 + (1-exp(-6*x))^6/6^2 +...
The series  B(x) = Sum_{n>=1} (1 - exp(-n*x))^n / n^2  may be rewritten as:
B(x) = Pi^2/6 + log(1-exp(-x)) + Sum_{n>=2} (n-1)*exp(-2*n*x)/(2*n) -
Sum_{n>=3} C(n-1,2)*exp(-3*n*x)/(3*n) + Sum_{n>=4} C(n-1,3)*exp(-4*n*x)/(4*n) -+...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
A. Ayyer, D. Hathcock and P. Tetali, Toppleable Permutations, Excedances and Acyclic Orientations, arXiv:2010.11236 [math.CO], 2020. For the precise definition of a toppleable permutation.

Cf. A136126, A092552, A048163, A220179, A122399, A187755, A203798, A320096, A338040.

Flatten[{1,Table[Sum[(-1)^(n-k)*k^n*k!*StirlingS2[n,k],{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Jun 21 2013 *)

{a(n)=polcoeff(sum(m=0,n,m^m*m!*x^m/prod(k=1,m,1+m*k*x+x*O(x^n))),n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=n!*polcoeff(sum(k=0, n, (1-exp(-k*x+x*O(x^n)))^k), n)}
for(n=0, 20, print1(a(n), ", "))

/* Formula for this sequence with offset=1: */
{a(n)=n!*polcoeff(sum(k=1, n, (1-exp(-k*x+x*O(x^n)))^k/k^2), n)}
for(n=1, 21, print1(a(n), ", "))

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = sum(k=0,n,(-1)^(n-k)*k^n*k!*Stirling2(n, k))}
for(n=0, 20, print1(a(n), ", "))

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = if(n==0,1,sum(k=1,n+1,((k-1)!)^2*Stirling2(n+1,k)^2/2))}
for(n=0, 20, print1(a(n), ", "))

{a(n)=sum(k=0,n, k^n*sum(j=0,k, (-1)^(n+k-j)*binomial(k,j)*(k-j)^n))}
for(n=0, 20, print1(a(n), ", "))

O.g.f. Sum_{n>=0} n^n * n! * x^n / Product_{k=1..n} (1 + n*k*x).
E.g.f. A(x) = Sum_{n>=0} (1 - exp(-n*x))^n  satisfies the identities:
(1) A(x) = Sum_{n>=1} exp(-n*x) * (1 - exp(-n*x))^(n-1).
(2) A(x) = 1 + (1/2) * Sum_{n>=1} (1 - exp(-n*x))^(n-1).
(3) A(x) = Sum_{n>=1} Sum_{k>=0} (-1)^k * C(n+k-1,k) * exp(-k*(n+k-1)*x).
E.g.f. at offset 1, B(x) = Sum_{n>=1} a(n-1)*x^n/n!, satisfies:
(1) B(x) = Sum_{n>=1} (1 - exp(-n*x))^n / n^2.
(2) B(x) = Pi^2/6 + log(1-exp(-x)) + Sum_{k>=2} Sum_{n>=k} (-1)^k * C(n-1,k-1) * exp(-k*n*x)/(k*n), a convergent series for x>0.
a(n) = Sum_{k=0..n} (-1)^(n-k) * k^n * k! * Stirling2(n,k).
a(n) = Sum_{k=1..n+1} ((k-1)!)^2 * Stirling2(n+1,k)^2 / 2 for n>0 with a(0)=1.
a(n) = Sum_{k=0..n} k^n * Sum_{j=0..k} (-1)^(n+k-j) * binomial(k,j) * (k-j)^n.
a(n) = A048163(n+1)/2 for n>0.
Limit n->infinity (a(n)/n!)^(1/n)/n = 1/(exp(1)*(log(2))^2) = 0.7656928576... - Vaclav Kotesovec, Jun 21 2013
a(n) ~ sqrt(Pi) * n^(2*n+1/2) / (sqrt(1-log(2)) * exp(2*n) * (log(2))^(2*n+1)). - Vaclav Kotesovec, May 13 2014

A306209 Number A(n,k) of permutations of [n] within distance k of a fixed permutation; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 6, 5, 1, 1, 1, 2, 6, 14, 8, 1, 1, 1, 2, 6, 24, 31, 13, 1, 1, 1, 2, 6, 24, 78, 73, 21, 1, 1, 1, 2, 6, 24, 120, 230, 172, 34, 1, 1, 1, 2, 6, 24, 120, 504, 675, 400, 55, 1, 1, 1, 2, 6, 24, 120, 720, 1902, 2069, 932, 89, 1, 1, 1, 2, 6, 24, 120, 720, 3720, 6902, 6404, 2177, 144, 1

Offset: 0

Alois P. Heinz, Jan 29 2019

A(n,k) counts permutations p of [n] such that |p(j)-j| <= k for all j in [n].

			A(4,1) = 5: 1234, 1243, 1324, 2134, 2143.
A(5,2) = 31: 12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 13425, 13524, 14235, 14253, 14325, 14523, 21345, 21354, 21435, 21453, 21534, 21543, 23145, 23154, 24135, 24153, 31245, 31254, 31425, 31524, 32145, 32154, 34125.
Square array A(n,k) begins:
  1,  1,   1,    1,    1,     1,     1,     1,     1, ...
  1,  1,   1,    1,    1,     1,     1,     1,     1, ...
  1,  2,   2,    2,    2,     2,     2,     2,     2, ...
  1,  3,   6,    6,    6,     6,     6,     6,     6, ...
  1,  5,  14,   24,   24,    24,    24,    24,    24, ...
  1,  8,  31,   78,  120,   120,   120,   120,   120, ...
  1, 13,  73,  230,  504,   720,   720,   720,   720, ...
  1, 21, 172,  675, 1902,  3720,  5040,  5040,  5040, ...
  1, 34, 400, 2069, 6902, 17304, 30960, 40320, 40320, ...

Alois P. Heinz, Antidiagonals n = 0..36, flattened
Torleiv Kløve, Spheres of Permutations under the Infinity Norm - Permutations with limited displacement, Reports in Informatics, Department of Informatics, University of Bergen, Norway, no. 376, November 2008.
Torleiv Kløve, Generating functions for the number of permutations with limited displacement, Electron. J. Combin., 16 (2009), #R104.

Columns k=0..10 give: A000012, A000045(n+1), A002524, A002526, A072856, A154654, A154655, A154656, A154657, A154658, A154659.
Rows n=1-2 give: A000012, A040000.
Main diagonal gives A000142.
A(2n,n) gives A048163(n+1).
A(2n+1,n) gives A092552(n+1).
A(n,floor(n/2)) gives A306267.
A(n+2,n) gives A001564.
Cf. A130152.

A[0, _] = 1;
A[n_ /; n > 0, k_] := A[n, k] = Permanent[Table[If[Abs[i - j] <= k, 1, 0], {i, 1, n}, {j, 1, n}]];
Table[A[n - k, k], {n, 0, 12}, {k, n, 0, -1 }] // Flatten (* Jean-François Alcover, Oct 18 2021, after Alois P. Heinz in A130152 *)

A(n,k) = Sum_{j=0..k} A130152(n,j) for n > 0, A(0,k) = 1.

A136126 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,k+n} having excedance set {1,2,...,k} (the empty set for k=0), 0 <= k <= n-1.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 31, 15, 1, 1, 31, 115, 115, 31, 1, 1, 63, 391, 675, 391, 63, 1, 1, 127, 1267, 3451, 3451, 1267, 127, 1, 1, 255, 3991, 16275, 25231, 16275, 3991, 255, 1, 1, 511, 12355, 72955, 164731, 164731, 72955, 12355, 511, 1

Offset: 1

Emeric Deutsch, Jan 17 2008

The excedance set of a permutation p in S_n is the set of indices i such that p(i) > i.
Columns 1,2,3,4 yield A000225, A091344, A091347, A091348, respectively. Row sums yield A136127.
T(a+b-1,b-1)*(-1)^(a+b-1) = Sum_{k=0..} F(a,b,k)*(-1)^k where F(a,b,k) is the number of connected subgraphs of K(a,b) (the complete bipartite graph) with k edges. F(n,n,k) is A255192(n,k). - Thomas Dybdahl Ahle, Feb 18 2015 [The sum starts with k=0, and F(n,n,k) is A255192(n,k), but there seems to be no A255192(n,0). Is there no upper k-summation limit? - Wolfdieter Lang, Mar 15 2015]
Comment from Don Knuth, Aug 25 2020, added by N. J. A. Sloane, Sep 07 2020: (Start)
This array also arises from a problem about {0,1}-matrices. Symmetric array read by antidiagonals: A(n,k) (n >= 1, k >= 0) = number of n X k matrices of 0's and 1's satisfying two conditions: (i) no column is entirely 0; (ii) no 0 has simultaneously a 1 above it and another 1 to its left.
Equivalently (see the Steingrímsson-Williams reference) A(n,k) is the number of permutations p_1...p_{n+k} on {1,...,n+k} for which p_1 >= 1, ..., p_n >= n, p_{n+1} < n+1,..., p_{n+k} < n+k. Then A(n,k) = A(k+1,n-1), for n >= 1 and k >= 0.
For example, the seven 2 X 2 matrices satisfying (i) and (ii) are
  00 01 10 10 11 11 11
  11 11 01 11 00 01 11
and the seven permutations of {1, 2, 3, 4} satisfying the other definition are
  1423, 2413, 3412, 3421, 4213, 4312, 4321.
(End)

			T(4,2) = 7 because 3412, 4312, 2413, 2314, 2431, 3421 and 4321 are the only permutations of {1,2,3,4} with excedance set {1,2}.
Triangle starts:
  1;
  1,   1;
  1,   3,    1;
  1,   7,    7,     1;
  1,  15,   31,    15,     1;
  1,  31,  115,   115,    31,     1;
  1,  63,  391,   675,   391,    63,    1;
  1, 127, 1267,  3451,  3451,  1267,  127,   1;
  1, 255, 3991, 16275, 25231, 16275, 3991, 255, 1;
  ...
Formatted as a square array A(n,k) with 0 <= k <= n:
  1,   1,    1,     1,      1,        1,         1,          1, ... [A000012]
  1,   3,    7,    15,     31,       63,       127,        255, ... [A000225]
  1,   7,   31,   115,    391,     1267,      3991,      12355, ... [A091344]
  1,  15,  115,   675,   3451,    16275,     72955,     316275, ... [A091347]
  1,  31,  391,  3451,  25231,   164731,    999391,    5767051, ... [A091348]
  1,  63, 1267, 16275, 164731,  1441923,  11467387,   85314915, ...
  1, 127, 3991, 72955, 999391, 11467387, 116914351, 1096832395, ...

Paul D. Hanna, Rows n = 1..31, flattened.
Tsuneo Arakawa and Masanobu Kaneko, Multiple zeta values, poly-Bernoulli numbers, and related zeta functions, Nagoya Math. J., 153:189-209, 1999.
Arvind Ayyer, Daniel Hathcock, and Prasad Tetali, Toppleable Permutations, Excedances and Acyclic Orientations, arXiv:2010.11236 [math.CO], 2020. Mentions this sequence.
Arvind Ayyer and Beáta Bényi, Toppling on permutations with an extra chip, arXiv:2104.13654 [math.CO], Apr. 2021. (See table 1.)
Beáta Bényi and Peter Hajnal, Combinatorial properties of poly-Bernoulli relatives, arXiv preprint arXiv:1602.08684 [math.CO], 2016. See C_{n,k}.
Beáta Bényi and Matthieu Josuat-Vergès, Combinatorial proof of an identity on Genocchi numbers, arXiv:2010.10060 [math.CO], 2020.
Taylor Brysiewicz and Aida Maraj, Lawrence Lifts, Matroids, and Maximum Likelihood Degrees, arXiv:2310.13064 [math.CO], 2023. See p. 13.
E. Clark and R. Ehrenborg, Explicit expressions for the extremal excedance statistic, European J. Combinatorics, 31, 2010, 270-279 (Theorem 3.1).
Bérénice Delcroix-Oger, Florent Hivert, Patxi Laborde-Zubieta, Jean-Christophe Aval, and Adrien Boussicault, Non-ambiguous trees: New results and generalisation (full version), arXiv:2103.07294 [cs.DM], 2021 (Proposition 1.8).
R. Ehrenborg and E. Steingrimsson, The excedance set of a permutation, Advances in Appl. Math., 24, 284-299, 2000 (Proposition 6.5).
Anatol N. Kirillov, On some quadratic algebras. I 1/2: Combinatorics of Dunkl and Gaudin elements, Schubert, Grothendieck, Fuss-Catalan, universal Tutte and reduced polynomials, SIGMA, Symmetry Integrability Geom. Methods Appl. 12, Paper 002, 172 p. (2016).
Don Knuth, Parades and poly-Bernoulli bijections, Mar 31 2024 (see Eq. (16.1)).
Toshiki Matsusaka, Symmetrized poly-Bernoulli numbers and combinatorics, arXiv:2003.12378 [math.NT], 2020. Table 1.
Einar Steingrímsson and Lauren K Williams, Permutation tableaux and permutation patterns, Journal of Combinatorial Theory, A 114 (2007), 211-234.
Julius Worpitzky, Studien über die Bernoullischen und Eulerschen Zahlen, Journal für die reine und angewandte Mathematik. 94:203-232, 1883.

Cf. A000225, A028246, A091344, A091347, A091348, A092552, A136127, A152884, A255192, A329369, A371761.

with(combinat): T:=proc(n,k) if k < n then add((-1)^(k+1-i)*factorial(i)*i^(n-1-k)* stirling2(k+1,i),i=1..k+1) else 0 end if end proc: for n to 10 do seq(T(n,k),k=0..n-1) end do; # yields sequence in triangular form
# Alternatively as a square array:
A := (n, k) -> add((-1)^(k-j)*j!*Stirling2(k+1,j+1)*(j+1)^(n+1), j=0..k);
seq(print(seq(A(n, k), k=0..7)), n=0..6); # Peter Luschny, Mar 14 2018
# Using the exponential generating function as given by Arakawa & Kaneko:
gf := polylog(-t, 1-exp(-x))/(exp(x)-1):
ser := series(gf, x, 12): c := n -> n!*coeff(ser, x, n):
seq(lprint(seq(subs(t=k, c(n)), n=0..8)), k=0..8); # Peter Luschny, Apr 29 2021
# Using recurrence relations:
A := proc(n, k) option remember; local j; if n = 0 then return k^n fi;
add(binomial(k+1, j+1)*A(n-1, k-j), j = 0..k) end:
for n from 0 to 7 do lprint(seq(A(n, k), k=0..8)) od;  # Peter Luschny, Apr 19 2024

T[n_, k_] := Sum[(-1)^(k + 1 - i)*i!*i^(n - 1 - k)*StirlingS2[k + 1, i], {i, 1, k + 1}];
Table[T[n, k], {n, 1, 10}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, Nov 16 2017 *)

{T(n,k)=polcoeff(polcoeff( x*y*sum(m=0, n, m!*x^m*prod(k=1, m, (1+y+k*x*y)/(1+(1+y)*k*x+k^2*x^2*y +x*O(x^n))) ), n,x),k,y)} \\ Paul D. Hanna, Feb 01 2013
for(n=1, 10,for(k=1,n, print1(T(n,k), ", "));print(""))

tabl(nn) = {default(seriesprecision, nn+1); pol = log(1/(1-(exp(x)-1)*(exp(y)-1))) + O(x^nn); for (n=1, nn-1, poly = n!*polcoeff(pol, n, x); for (k=1, n, print1(k!*polcoeff(poly, k, y), ", ");); print(););} \\ Michel Marcus, Apr 17 2015

T(n,k) = Sum_{i=1..k+1} (-1)^(k+1-i)*i!*i^(n-1-k)*Stirling2(k+1,i) (0 <= k <= n-1).
G.f.: A(x,y) = x*y*Sum_{n>=1} n! * x^n*Product_{k=1..n} (1 + y + k*x*y) / (1 + (1+y)*k*x + k^2*x^2*y). - Paul D. Hanna, Feb 01 2013
Central terms of triangle equals A092552. - Paul D. Hanna, Feb 01 2013
T(n,k-1) = Sum_{i=0..k, m=0..i} binomial(i,m)*(-1)^(k-m)*i^(n-k)*m^k (1 <= k <= n). - Thomas Dybdahl Ahle, Feb 18 2015
E.g.f.: log(1/(1-(exp(x)-1)*(exp(y)-1))). - Vladimir Kruchinin, Apr 17 2015
Let W(n,k) = k!*Stirling2(n+1, k+1) denote the Worpitzky numbers, then A(n,k) = Sum_{j=0..k} (-1)^(k-j)*W(k,j)*(j+1)^(n+1) enumerates the square array. - Peter Luschny, Mar 14 2018
Assume the missing first row (1,0,0,...) of the array which Ayyer and Bényi call  the 'poly-Bernoulli numbers of type C'. Then T(n, k) = p_{n}(k) where p_{n}(x) = Sum_{k=0..n} (-1)^(n-k)*(k+1)^x*Sum_{j=0..n} E1(n,j)*binomial(n-j, n-k), and E1(n, k) are the Eulerian numbers of first order. This reflects the Worpitzky approach to the Bernoulli numbers. This formula can alternatively be written as: T(n, k) = Sum_{j=0..k} (-1)^(k-j)*(j+1)^n*A028246(k+1, j+1). - Peter Luschny, Apr 29 2021

Definition corrected. Changed "T(n,k) is the number of permutations of {1,2,...,n}..." to "T(n,k) is the number of permutations of {1,2,...,k+n}..." - Karel Casteels (kcasteel(AT)sfu.ca), Feb 17 2010

A244585 E.g.f.: Sum_{n>=1} (exp(n*x) - 1)^n / n.

Original entry on oeis.org

1, 5, 79, 2621, 149071, 12954365, 1596620719, 264914218301, 56934521042191, 15385666763366525, 5106110041462786159, 2041611328770984737981, 967972254733121945653711, 536962084044317668770841085, 344546100916295014902350596399

Offset: 1

Paul D. Hanna, Aug 21 2014

nonn

Compare to: Sum_{n>=1} (1 - exp(-n*x))^n / n, the e.g.f. of A092552.

			E.g.f.: A(x) = x + 5*x^2/2! + 79*x^3/3! + 2621*x^4/4! + 149071*x^5/5! +...
where
A(x) = (exp(x)-1) + (exp(2*x)-1)^2/2 + (exp(3*x)-1)^3/3 + (exp(4*x)-1)^4/4 + (exp(5*x)-1)^5/5 + (exp(6*x)-1)^6/6 + (exp(7*x)-1)^7/7 +...
Exponentiation yields:
exp(A(x)) = 1 + x + 6*x^2/2! + 95*x^3/3! + 3043*x^4/4! + 167342*x^5/5! +...+ A243802(n)*x^n/n! +...
The O.G.F. begins:
F(x) = x + 5*x^2 + 79*x^3 + 2621*x^4 + 149071*x^5 + 12954365*x^6 +...
where
F(x) = x/(1-x) + 2*2!*x^2/((1-2*x)*(1-4*x)) + 3^2*3!*x^3/((1-3*x)*(1-6*x)*(1-9*x)) + 4^3*4!*x^4/((1-4*x)*(1-8*x)*(1-12*x)*(1-16*x)) + 5^4*5!*x^5/((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)) +...

Vaclav Kotesovec, Table of n, a(n) for n = 1..200

Cf. A092552, A243802.

{a(n) = n!*polcoeff( sum(m=1,n+1, (exp(m*x +x*O(x^n)) - 1)^m / m), n)}
for(n=0,20,print1(a(n),", "))

{a(n)=if(n<1, 0, polcoeff(sum(m=1, n, m^(m-1) * m! * x^m / prod(k=1, m, 1-m*k*x +x*O(x^n))), n))}
for(n=0, 20, print1(a(n), ", "))

O.g.f.: Sum_{n>=1} n^(n-1) * n! * x^n / Product_{k=1..n} (1 - n*k*x).

a(n) ~ c * d^n * (n!)^2 / n^(3/2), where d = A317855 = (1+exp(1/r))*r^2 = 3.161088653865428813830172202588132491..., r = 0.873702433239668330496568304720719298... is the root of the equation exp(1/r)/r + (1+exp(1/r)) * LambertW(-exp(-1/r)/r) = 0, and c = 0.37498840921734807101035131780130551... . - Vaclav Kotesovec, Aug 21 2014

A212084 Triangle T(n,k), n>=0, 0<=k<=2n, read by rows: row n gives the coefficients of the chromatic polynomial of the complete bipartite graph K_(n,n), highest powers first.

Original entry on oeis.org

1, 1, -1, 0, 1, -4, 6, -3, 0, 1, -9, 36, -75, 78, -31, 0, 1, -16, 120, -524, 1400, -2236, 1930, -675, 0, 1, -25, 300, -2200, 10650, -34730, 75170, -102545, 78610, -25231, 0, 1, -36, 630, -6915, 52080, -279142, 1074822, -2942445, 5552680, -6796926, 4787174

Offset: 0

Alois P. Heinz, Apr 30 2012

The complete bipartite graph K_(n,n) has 2n vertices and n^2 = A000290(n) edges. The chromatic polynomial of K_(n,n) has 2n+1 = A005408(n) coefficients.

			3 example graphs:                     +-----------+
.                 o        o   o      o   o   o   |
.                 |        |\ /|      |\ /|\ /|\ /
.                 |        | X |      | X | X | X
.                 |        |/ \|      |/ \|/ \|/ \
.                 o        o   o      o   o   o   |
.                                     +-----------+
Graph:         K_(1,1)    K_(2,2)      K_(3,3)
Vertices:         2          4            6
Edges:            1          4            9
The complete bipartite graph K_(2,2) is the cycle graph C_4 with chromatic polynomial q^4 -4*q^3 +6*q^2 -3*q => row 2 = [1, -4, 6, -3, 0].
Triangle T(n,k) begins:
  1;
  1,  -1,   0;
  1,  -4,   6,    -3,     0;
  1,  -9,  36,   -75,    78,     -31,       0;
  1, -16, 120,  -524,  1400,   -2236,    1930,     -675, ...
  1, -25, 300, -2200, 10650,  -34730,   75170,  -102545, ...
  1, -36, 630, -6915, 52080, -279142, 1074822, -2942445, ...
  ...

Alois P. Heinz, Rows n = 0..90, flattened
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Wikipedia, Chromatic Polynomial

Columns k=0-2 give: A000012, (-1)*A000290, A083374.
Row sums and last elements of rows give: A000007.
Row lengths give: A005408.
Sums of absolute values of row elements give: A048163(n+1).
T(n,2n-1) = (-1)*A092552(n).
Cf. A008277, A212085, A182368, A185442, A193233, A193277, A193283, A266695, A266972.

P:= n-> add(Stirling2(n, k) *mul(q-i, i=0..k-1) *(q-k)^n, k=0..n):
T:= n-> seq(coeff(P(n), q, 2*n-k), k=0..2*n):
seq(T(n), n=1..8);

T(n,k) = [q^(2n-k)] Sum_{j=0..n} (q-j)^n * S2(n,j) * Product_{i=0..j-1} (q-i).

T(0,0)=1 prepended by Alois P. Heinz, May 03 2024

A220179 E.g.f.: Sum_{n>=1} (1 - exp(-n^2*x))^n / n.

Original entry on oeis.org

1, 15, 1267, 316275, 174397531, 179770837155, 310789895286907, 834906367019076675, 3293344593080631993211, 18259284528276047000517795, 137429981152689382429349060347, 1365009985652048448232840864764675, 17475885712645599218827214639383437691

Offset: 1

Paul D. Hanna, Dec 06 2012

nonn

Compare to the trivial identity: x = Sum_{n>=1} (1 - exp(-x))^n/n.

Compare to the e.g.f. of A092552: Sum_{n>=1} (1 - exp(-n*x))^n/n.

			E.g.f.: A(x) = x + 15*x^2/2! + 1267*x^3/3! + 316275*x^4/4! + 174397531*x^5/5! +...
where
A(x) = (1-exp(-x)) + (1-exp(-4*x))^2/2 + (1-exp(-9*x))^3/3 + (1-exp(-16*x))^4/4 + (1-exp(-25*x))^5/5 +...

Seiichi Manyama, Table of n, a(n) for n = 1..162

Cf. A092552, A187755, A220181, A242228.

Table[Sum[(-1)^(n-k) * k^(2*n-1) * k! * StirlingS2[n,k], {k,1,n}], {n,1,20}] (* Vaclav Kotesovec, May 08 2014 *)

a(n)=n!*polcoeff(sum(k=1, n, (1-exp(-k^2*x+x*O(x^n)))^k/k), n)
for(n=1,20,print1(a(n),", "))

a(n)=polcoeff(sum(m=1, n, m^(2*m-1)*m!*x^m/prod(k=1, m, 1+m^2*k*x+x*O(x^n))), n) \\ Paul D. Hanna, Jan 05 2013
for(n=1,20,print1(a(n),", "))

{a(n)=sum(k=1, n, (-1)^(n-k)*k^(2*n-1)*k!*stirling(n, k, 2))}
for(n=1, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Jan 05 2013

O.g.f.: Sum_{n>=1} n^(2*n-1) * n! * x^n / Product_{k=1..n} (1 - n^2*k*x). - Paul D. Hanna, Jan 05 2013
a(n) = Sum_{k=1..n} (-1)^(n-k) * k^(2*n-1) * k! * Stirling2(n,k). - Paul D. Hanna, Jan 05 2013
a(n) ~ c * d^n * (n!)^3 / n^2, where d = 6.8312860494079582446988970296645779575650627187418208311407895492635... and c = 0.175744118254830086361220160145768507562830495967... . - Vaclav Kotesovec, May 08 2014

A188634 E.g.f.: Sum_{n>=0} (1 - exp(-(n+1)*x))^n/(n+1).

Original entry on oeis.org

1, 1, 4, 46, 1066, 41506, 2441314, 202229266, 22447207906, 3216941445106, 578333776748674, 127464417117501586, 33800841048945424546, 10617398393395844992306, 3898852051843774954576834, 1654948033478889053351543506, 804119629083230641164978005986

Offset: 0

Paul D. Hanna, Dec 28 2012

nonn

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 46*x^3/3! + 1066*x^4/4! + 41506*x^5/5! +...
where
A(x) = 1 + (1-exp(-2*x))/2 + (1-exp(-3*x))^2/3 + (1-exp(-4*x))^3/4 + (1-exp(-5*x))^4/5 + (1-exp(-6*x))^5/6 +...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A099594, A092552, A191908, A266695.

Table[Sum[(-1)^(k+n)*(k+1)^(n-1)*k!*StirlingS2[n, k],{k,0,n}],{n,0,20}]
Table[n!*SeriesCoefficient[Sum[(1-E^(-x*(k+1)))^k/(k+1),{k,0,n}],{x,0,n}],{n,0,20}]  (* Vaclav Kotesovec, Dec 30 2012 *)

{a(n)=n!*polcoeff(sum(k=0, n, (1-exp(-(k+1)*x+x*O(x^n)))^k/(k+1)), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=sum(j=0,n, (j+1)^(n-1)*sum(i=0,j, (-1)^(n+j-i)*binomial(j,i)*(j-i)^n))}
for(n=0, 20, print1(a(n), ", "))

a(n) = Sum_{j=0..n} (j+1)^(n-1) * Sum_{i=0..j} (-1)^(n+j-i)*C(j, i)*(j-i)^n.
Ignoring the initial term, equals a diagonal of array A099594, which forms the poly-Bernoulli numbers B(-k,n).
Limit n->infinity a(n)^(1/n)/n^2 = 0.281682... - Vaclav Kotesovec, Dec 30 2012
a(n) = A266695(2*n-1) for n >= 1. - Alois P. Heinz, Apr 17 2024

cp's OEIS Frontend

A220181 E.g.f.: Sum_{n>=0} (1 - exp(-n*x))^n.

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A306209 Number A(n,k) of permutations of [n] within distance k of a fixed permutation; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A136126 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,k+n} having excedance set {1,2,...,k} (the empty set for k=0), 0 <= k <= n-1.

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A244585 E.g.f.: Sum_{n>=1} (exp(n*x) - 1)^n / n.

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A212084 Triangle T(n,k), n>=0, 0<=k<=2n, read by rows: row n gives the coefficients of the chromatic polynomial of the complete bipartite graph K_(n,n), highest powers first.

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A220179 E.g.f.: Sum_{n>=1} (1 - exp(-n^2*x))^n / n.

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