A001861 -id:A001861 - cp's OEIS frontend

A001883 Number of permutations s of {1,2,...,n} such that |s(i)-i|>1 for each i=1,2,...,n.

1, 0, 0, 0, 1, 4, 29, 206, 1708, 15702, 159737, 1780696, 21599745, 283294740, 3995630216, 60312696452, 970234088153, 16571597074140, 299518677455165, 5711583170669554, 114601867572247060, 2413623459384988298, 53238503492701261201, 1227382998752177970288, 29520591675204638641249

Offset: 0

N. J. A. Sloane

nonn

Permanent of the (0,1)-matrix having (i,j)-th entry equal to 0 iff this is on the first lower-diagonal, diagonal or the first upper-diagonal. - Simone Severini, Oct 14 2004

J. Riordan, "The enumeration of permutations with three-ply staircase restrictions," unpublished memorandum, Bell Telephone Laboratories, Murray Hill, NJ, Oct 1963.
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).

Eric M. Schmidt, Table of n, a(n) for n = 0..200
Dmitry Efimov, Determinants of generalized binary band matrices, arXiv:1702.05655 [math.RA], 2017.
J. Riordan, Letter, Oct 31 1977
N. J. A. Sloane, Annotated copy of Riordan's Three-Ply Staircase paper
Donovan Young, The Number of Domino Matchings in the Game of Memory, J. Int. Seq., Vol. 21 (2018), Article 18.8.1.
Donovan Young, Generating Functions for Domino Matchings in the 2 X k Game of Memory, J. Int. Seq., Vol. 22 (2019), Article 19.8.7.
D. Zeilberger, Automatic Enumeration of Generalized Menage Numbers, arXiv preprint arXiv:1401.1089 [math.CO], 2014.

Cf. A001884-A001891, A075851, A075852.
Also a diagonal of A080018.
Column k=0 of A323671.

b:= proc(n, s) option remember; `if`(n=0, 1, add(
      `if`(abs(n-i)<=1, 0, b(n-1, s minus {i})), i=s))
    end:
a:= n-> b(n, {$1..n}):
seq(a(n), n=0..15);  # Alois P. Heinz, Jul 04 2015

b[n_, s_List] := b[n, s] = If[n == 0, 1, Sum[If[Abs[n-i] <= 1, 0, b[n-1, s ~Complement~ {i}]], {i, s}]]; a[n_] := b[n, Range[n]]; Table[Print[a[n]]; a[n], {n, 4, 24}] (* Jean-François Alcover, Nov 10 2015, after Alois P. Heinz *)

permRWNb(a)=n=matsize(a)[1]; if(n==1,return(a[1,1])); sg=1; in=vectorv(n); x=in; x=a[,n]-sum(j=1,n,a[,j])/2; p=prod(i=1,n,x[i]); for(k=1,2^(n-1)-1,sg=-sg; j=valuation(k,2)+1; z=1-2*in[j]; in[j]+=z; x+=z*a[,j]; p+=prod(i=1,n,x[i],sg)); return(2*(2*(n%2)-1)*p)
for(n=1,23,a=matrix(n,n,i,j,abs(i-j)>1);print1(permRWNb(a)",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 16 2007

a(n) = (n+1)*a(n-1) - (n-3)*a(n-2) - (n-4)*a(n-3) + (n-4)*a(n-4) + a(n-5) + (-1)^n * Lucas(n-3), n > 4. [Riordan] (Note: There is a slight mistake in Riordan's paper. On p. 3 it should say that a_5 = 3.) - Eric M. Schmidt, Oct 09 2017
From Vaclav Kotesovec, Oct 10 2017: (Start)
a(n) = n*a(n-1) + 4*a(n-2) - 3*(n-3)*a(n-3) + (n-4)*a(n-4) + 2*(n-5)*a(n-5) - (n-7)*a(n-6) - a(n-7).
a(n) ~ exp(-3) * n!.
(End)

More terms and better description from Reiner Martin, Oct 14 2002
More terms from Vladimir Baltic, Vladeta Jovovic, Jan 04 2003
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 16 2007
a(22)-a(24) from Alois P. Heinz, Jul 04 2015
a(0)-a(3) from Eric M. Schmidt, Oct 09 2017

A005494 3-Bell numbers: E.g.f.: exp(3*z + exp(z) - 1).

Original entry on oeis.org

1, 4, 17, 77, 372, 1915, 10481, 60814, 372939, 2409837, 16360786, 116393205, 865549453, 6713065156, 54190360453, 454442481041, 3952241526188, 35590085232519, 331362825860749, 3185554606447814, 31581598272055879, 322516283206446897, 3389017736055752178

Offset: 0

N. J. A. Sloane, Simon Plouffe

nonn

For further information, references, programs, etc. for r-Bell numbers see A005493. - N. J. A. Sloane, Nov 27 2013
From expansion of falling factorials (binomial transform of A005493).
Row sums of Sheffer triangle (exp(3*x), exp(x)-1). - Wolfdieter Lang, Sep 29 2011

			G.f. = 1 + 4*x + 17*x^2 + 77*x^3 + 372*x^4 + 1915*x^5 + 10481*x^6 + 60814*x^7 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Eldar Fischer, Johann A. Makowsky, and Vsevolod Rakita, MC-finiteness of restricted set partition functions, arXiv:2302.08265 [math.CO], 2023.
R. Jakimczuk, Successive Derivatives and Integer Sequences, J. Int. Seq. 14 (2011) # 11.7.3.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
Toufik Mansour and Mark Shattuck, A recurrence related to the Bell numbers, INTEGERS 11 (2011), #A67.
I. Mezo, The r-Bell numbers, J. Int. Seq. 14 (2011) # 11.1.1.
J. Riordan, Letter, Oct 31 1977
N. J. A. Sloane, Transforms
Earl Glen Whitehead Jr., Stirling number identities from chromatic polynomials, J. Combin. Theory, A 24 (1978), 314-317.

Cf. A000110, A005493, A045379, A108087, A124323, A196834.

A row or column of the array A108087.

A005494:= func< n | (&+[Binomial(n,j)*3^(n-j)*Bell(j): j in [0..n]]) >;
[A005494(n): n in [0..30]]; // G. C. Greubel, Dec 01 2022

seq(add(3^(n-i)*combinat:-bell(i)*binomial(n,i),i=0..n), n=0..50); # Robert Israel, Dec 16 2014
# second Maple program:
b:= proc(n, m) option remember; `if`(n=0,
      m^2, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n+1, 0)-b(n, 0):
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 03 2025

Range[0, 40]! CoefficientList[Series[Exp[3 x + Exp[x] - 1], {x, 0, 40}], x] (* Vincenzo Librandi, Mar 04 2014 *)

def A005494(n): return sum( 3^(n-j)*bell_number(j)*binomial(n,j) for j in range(n+1))
[A005494(n) for n in range(31)] # G. C. Greubel, Dec 01 2022

a(n) = Sum_{i=0..n} 3^(n-i)*B(i)*binomial(n,i) where B(n) = Bell numbers A000110(n). - Fred Lunnon, Aug 04 2007
a(n) = exp(-1)*Sum_{k>=0} ((k+3)^n)/k!. - Gerald McGarvey, Jun 03 2004. May be rewritten as a(n) = Sum_{k>=3} (k^n*(k-1)*(k-2)/k!)/exp(1), which is a Dobinski-type relation for this sequence. - Karol A. Penson, Aug 18 2006
Define f_1(x), f_2(x), ... such that f_1(x) = x^2*e^x, f_{n+1}(x) = (d/dx)(x*f_n(x)), for n=2,3,.... Then a(n-1) = e^(-1)*f_n(1). - Milan Janjic, May 30 2008
Let A be the upper Hessenberg matrix of order n defined by: A[i,i-1]=-1, A[i,j]=binomial(j-1,i-1), (i <= j), and A[i,j]=0 otherwise. Then, for n >= 1, a(n) = (-1)^(n)charpoly(A,-3). - Milan Janjic, Jul 08 2010
a(n) = Sum_{k=3..n+3} A143495(n+3,k), n >= 0. - Wolfdieter Lang, Sep 29 2011
G.f.: 1/U(0) where U(k)= 1 - x*(k+4) - x^2*(k+1)/U(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 11 2012
G.f.: Sum_{k>0} x^(k-1) / ((1 - 3*x) * (1 - 4*x) * ... * (1 - (k+2)*x)). - Michael Somos, Feb 26 2014
G.f.: Sum_{k>0} k * x^(k-1) / ((1 - 2*x) * (1 - 3*x) * ... * (1 - (k+1)*x)). - Michael Somos, Feb 26 2014
a(n) ~ exp(n/LambertW(n) - n - 1) * n^(n + 3) / LambertW(n)^(n + 7/2). - Vaclav Kotesovec, Jun 10 2020
a(0) = 1; a(n) = 3 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k). - Ilya Gutkovskiy, Jul 02 2020
a(n) = Sum_{k=0..n} 4^k*A124323(n, k). - Mélika Tebni, Jun 10 2022

A109128 Triangle read by rows: T(n,k) = T(n-1,k-1) + T(n-1,k) + 1 for 0

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 11, 7, 1, 1, 9, 19, 19, 9, 1, 1, 11, 29, 39, 29, 11, 1, 1, 13, 41, 69, 69, 41, 13, 1, 1, 15, 55, 111, 139, 111, 55, 15, 1, 1, 17, 71, 167, 251, 251, 167, 71, 17, 1, 1, 19, 89, 239, 419, 503, 419, 239, 89, 19, 1, 1, 21, 109, 329, 659, 923, 923, 659, 329, 109, 21, 1

Offset: 0

Reinhard Zumkeller, Jun 20 2005

Eigensequence of the triangle = A001861. - Gary W. Adamson, Apr 17 2009

			Triangle begins as:
  1;
  1   1;
  1   3   1;
  1   5   5   1;
  1   7  11   7   1;
  1   9  19  19   9   1;
  1  11  29  39  29  11   1;
  1  13  41  69  69  41  13   1;
  1  15  55 111 139 111  55  15   1;
  1  17  71 167 251 251 167  71  17   1;
  1  19  89 239 419 503 419 239  89  19   1;

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened
Index entries for triangles and arrays related to Pascal's triangle

Cf. A000035, A000295, A001861, A005408, A014473, A028387, A030662.
Cf. A134760, A281362, A327767.
Cf. A000325 (row sums).
Sequence m*binomial(n,k) - (m-1): A007318 (m=1), this sequence (m=2), A131060 (m=3), A131061 (m=4), A131063 (m=5), A131065 (m=6), A131067 (m=7), A168625 (m=8).

a109128 n k = a109128_tabl !! n !! k
a109128_row n = a109128_tabl !! n
a109128_tabl = iterate (\row -> zipWith (+)
   ([0] ++ row) (1 : (map (+ 1) $ tail row) ++ [0])) [1]
-- Reinhard Zumkeller, Apr 10 2012

[2*Binomial(n,k) -1: k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 12 2020

A109128 := proc(n,k)
    2*binomial(n,k)-1 ;
end proc: # R. J. Mathar, Jul 12 2016

Table[2*Binomial[n,k] -1, {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 12 2020 *)

[[2*binomial(n,k) -1 for k in (0..n)] for n in (0..12)] # G. C. Greubel, Mar 12 2020

T(n,k) = T(n-1,k-1) + T(n-1,k) + 1 with T(n,0) = T(n,n) = 1.
Sum_{k=0..n} T(n, k) = A000325(n+1) (row sums).
T(n, k) = 2*binomial(n,k) - 1. - David W. Cantrell (DWCantrell(AT)sigmaxi.net), Sep 30 2007
T(n, 1) = 2*n - 1 = A005408(n+1) for n>0.
T(n, 2) = n^2 + n - 1 = A028387(n-2) for n>1.
T(n, k) = Sum_{j=0..n-k} C(n-k,j)*C(k,j)*(2 - 0^j) for k <= n. - Paul Barry, Apr 27 2006
T(n,k) = A014473(n,k) + A007318(n,k), 0 <= k <= n. - Reinhard Zumkeller, Apr 10 2012
From G. C. Greubel, Apr 06 2024: (Start)
T(n, n-k) = T(n, k).
T(2*n, n) = A134760(n).
T(2*n-1, n) = A030662(n), for n >= 1.
Sum_{k=0..n-1} T(n, k) = A000295(n+1), for n >= 1.
Sum_{k=0..n} (-1)^k*T(n, k) = 2*[n=0] - A000035(n+1).
Sum_{k=0..n-1} (-1)^k*T(n, k) = A327767(n), for n >= 1.
Sum_{k=0..floor(n/2)} T(n-k, k) = A281362(n).
Sum_{k=0..floor((n-1)/2)} T(n-k, k) = A281362(n-1) - (1+(-1)^n)/2 for n >= 1.
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = b(n), where b(n) is the repeating pattern {1,1,0,-2,-3,-1,2,2,-1,-3,-2,0} with b(n) = b(n-12). (End)

Offset corrected by Reinhard Zumkeller, Apr 10 2012

A035009 STIRLING transform of [1,1,2,4,8,16,32,...].

Original entry on oeis.org

1, 1, 3, 11, 47, 227, 1215, 7107, 44959, 305091, 2206399, 16913987, 136823263, 1163490499, 10366252031, 96491364675, 935976996127, 9440144423875, 98800604237119, 1071092025420867, 12008090971866207, 139014305916844739, 1659578039401022079, 20405708646650507075

Offset: 0

N. J. A. Sloane

Numerators of sequence that shifts left one place under 1/2 order binomial transform. (Denominators are 2^(n-1) for n > 0.) - Franklin T. Adams-Watters, Jul 31 2005
Row sums of triangle A137597 starting (1, 3, 11, 47, 227, ...). - Gary W. Adamson, Jan 29 2008
From Gary W. Adamson, Jul 22 2011: (Start)
a(n)/2^(n-1) = upper left term in M^n, M = an infinite square production matrix in which a column of (1/2, 1/2, 1/2, ...) is appended to the right of Pascal's triangle, as follows:
  1,  1/2,  0,   0,   0,   0,  ...
  1,   1,  1/2,  0,   0,   0,  ...
  1,   2,   1,  1/2,  0,   0,  ...
  1,   3,   3,   1,  1/2,  0,  ...
  1,   4,   6,   4,   1,  1/2, ..., etc.
(End)
From Bruno Berselli, Mar 20 2013: (Start)
Note that, for t=A222391:
a(1)*t = Sum_{n >= 1} 1  /(Gamma(n/2)*Gamma((n+1)/2)),
a(2)*t = Sum_{n >= 1} n  /(Gamma(n/2)*Gamma((n+1)/2)),
a(3)*t = Sum_{n >= 1} n^2/(Gamma(n/2)*Gamma((n+1)/2)),
a(4)*t = Sum_{n >= 1} n^3/(Gamma(n/2)*Gamma((n+1)/2)),
a(5)*t = Sum_{n >= 1} n^4/(Gamma(n/2)*Gamma((n+1)/2)),
a(6)*t = Sum_{n >= 1} n^5/(Gamma(n/2)*Gamma((n+1)/2)), etc.
(End)
Except for the initial term, the main diagonal of A129340. - Peter Bala, Apr 14 2017

			Given the production matrix M, upper left term of M^5 = a(5)/2^4 = 227/16.

Seiichi Manyama, Table of n, a(n) for n = 0..558
Paul Barry, Eulerian-Dowling Polynomials as Moments, Using Riordan Arrays, arXiv:1702.04007 [math.CO], 2017.
Mikhail Khovanov, Victor Ostrik and Yakov Kononov, Two-dimensional topological theories, rational functions and their tensor envelopes, arXiv:2011.14758 [math.QA], 2020.
Toufik Mansour and Mark Shattuck, A recurrence related to the Bell Numbers, Integers 11 (2011), #A67.

Cf. A000110, A011782, A137597, A222391, A129340.

A035009 := proc(n) local a,b,i;
a := [seq(2,i=1..n-1)]; b := [seq(1,i=1..n-1)];
exp(-x)*hypergeom(a,b,x); round(evalf(subs(x=2,%), 10+2*n)) end:
seq(A035009(n),n=0..19);  # Peter Luschny, Mar 30 2011
# second Maple program:
b:= proc(n, m) option remember;
     `if`(n=0, ceil(2^(m-1)), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 03 2021

1/(2*E^2)*Sum[(i + j)^n/(i!*j!), {i, 0, Infinity}, {j, 0, Infinity}] (* Starting from the 2nd term *) (* Vladimir Reshetnikov, Dec 31 2008 *)
Join[{1}, Table[BellB[n, 2]/2, {n, 1, 25}]] (* Vaclav Kotesovec, Jun 26 2022 *)

x='x+O('x^99); Vec(serlaplace((1 + exp(2*exp(x)-2))/2)) \\ Joerg Arndt, Apr 01 2011

a(n) = (1/2)*A001861(n), n > 0.
E.g.f.: (1 + exp(2*exp(x)-2))/2. - Emeric Deutsch, Feb 09 2002
a(n+1) = 1 + 2*Sum_{j=1..n} binomial(n, j)*a(j). - Jon Perry, Apr 25 2005
Define f_1(x), f_2(x), ... such that f_1(x)=e^x and for n=2,3,... f_{n+1}(x) = (d/dx)(x*f_n(x)). Then a(n) = e^(-2)*f_n(2). - Milan Janjic, May 30 2008
G.f.: 1 + x/(Q(0) - 2*x) where Q(k) =  1 - x*(k+1)/( 1 - 2*x/Q(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Mar 22 2013
G.f.: 1/Q(0), where Q(k)= 1 - x - 2*x/(1 - x*(2*k+1)/(1 - x - 2*x/(1 - x*(2*k+2)/Q(k+1)))); (continued fraction). - Sergei N. Gladkovskii, May 13 2013
G.f.: 1 + Sum_{k>=1} 2^(k-1)*x^k/Product_{j=1..k} (1 - j*x). - Ilya Gutkovskiy, Jun 19 2018

A049020 Triangle of numbers a(n,k), 0 <= k <= n: number of set partitions of {1,2,...,n} in which exactly k of the blocks have been distinguished.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 5, 10, 6, 1, 15, 37, 31, 10, 1, 52, 151, 160, 75, 15, 1, 203, 674, 856, 520, 155, 21, 1, 877, 3263, 4802, 3556, 1400, 287, 28, 1, 4140, 17007, 28337, 24626, 11991, 3290, 490, 36, 1, 21147, 94828, 175896, 174805, 101031, 34671, 6972, 786, 45, 1

Offset: 0

N. J. A. Sloane

Triangle a(n,k) read by rows; given by [1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is Deléham's operator defined in A084938.
Exponential Riordan array [exp(exp(x)-1), exp(x)-1]. - Paul Barry, Jan 12 2009
Equal to A048993*A007318. - Philippe Deléham, Oct 31 2011
This lower unitriangular array is the L factor in the LU decomposition of the Hankel matrix (Bell(i+j-2))A000110(n).%20The%20U%20factor%20is%20A059098%20(see%20Chamberland,%20p.%201672).%20-%20_Peter%20Bala">i,j >= 1, where Bell(n) = A000110(n). The U factor is A059098 (see Chamberland, p. 1672). - _Peter Bala, Oct 15 2023

			Triangle begins:
   1;
   1,  1;
   2,  3,  1;
   5, 10,  6,  1;
  15, 37, 31, 10,  1;
  ...
From _Paul Barry_, Jan 12 2009: (Start)
Production array begins
  1, 1;
  1, 2, 1;
  0, 2, 3, 1;
  0, 0, 3, 4, 1;
  0, 0, 0, 4, 5, 1;
  ... (End)

Alois P. Heinz, Rows n = 0..140, flattened
M. Aigner, A characterization of the Bell numbers, Discr. Math., 205 (1999), 207-210.
Paul Barry, Constructing Exponential Riordan Arrays from Their A and Z Sequences, Journal of Integer Sequences, 17 (2014), #14.2.6.
Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, Combinatorial Identities for Vacillating Tableaux, arXiv:2308.14183 [math.CO], 2023. See p. 11.
Marc Chamberland, Factored matrices can generate combinatorial identities, Linear Algebra and its Applications, Volume 438, Issue 4, 15 Feb. 2013, pp. 1667-1677.
J. East and R. D. Gray, Idempotent generators in finite partition monoids and related semigroups, arXiv preprint arXiv:1404.2359 [math.GR], 2014.
Tom Halverson and Theodore N. Jacobson, Set-partition tableaux and representations of diagram algebras, arXiv:1808.08118 [math.RT], 2018.
Aoife Hennessy and Paul Barry, Generalized Stirling Numbers, Exponential Riordan Arrays, and Orthogonal Polynomials, J. Int. Seq. 14 (2011) # 11.8.2.
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
W. F. Lunnon et al., Arithmetic properties of Bell numbers to a composite modulus I, Acta Arith., 35 (1979), 1-16.
J. Riordan, Letter, Oct 31 1977. The array is on the second page.

First column gives A000110, second column = A005493.
Third column = A003128, row sums = A001861, A059340.
See A244489 for another version of this triangle.
Cf. A059098, A245109, A046716.

a:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
      `if`(n=0, 1, a(n-1, k-1) +(k+1)*(a(n-1, k) +a(n-1, k+1))))
    end:
seq(seq(a(n, k), k=0..n), n=0..15);  # Alois P. Heinz, Nov 30 2012

a[n_, k_] = Sum[StirlingS2[n, i]*Binomial[i, k], {i, 0, n}]; Flatten[Table[a[n, k], {n, 0, 9}, {k, 0, n}]]
(* Jean-François Alcover, Aug 29 2011, after Vladeta Jovovic *)

T(n,k)=if(k<0 || k>n,0,n!*polcoeff(polcoeff(exp((1+y)*(exp(x+x*O(x^n))-1)),n),k))

def A049020_triangle(dim):
    M = matrix(ZZ, dim, dim)
    for n in (0..dim-1): M[n, n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            M[n, k] = M[n-1, k-1]+(k+1)*M[n-1, k]+(k+1)*M[n-1, k+1]
    return M
A049020_triangle(9) # Peter Luschny, Sep 19 2012

a(n,k) = a(n-1, k-1) + (k+1)*a(n-1, k) + (k+1)*a(n-1, k+1), n >= 1.
a(n,k) = Sum_{i=0..n} Stirling2(n,i)*binomial(i,k). - Vladeta Jovovic, Jan 27 2001
E.g.f. for the k-th column is (1/k!)*(exp(x)-1)^k*exp(exp(x)-1). - Vladeta Jovovic, Jan 27 2001
G.f.: 1/(1-x-xy-x^2(1+y)/(1-2x-xy-2x^2(1+y)/(1-3x-xy-3x^2(1+y)/(1-4x-xy-4x^2(1+y)/(1-... (continued fraction). - Paul Barry, Apr 29 2009
E.g.f.: exp((y+1)*(exp(x)-1)). - Geoffrey Critzer, Nov 30 2012
Note that A244489 (which is essentially the same triangle) has the formula T(n,k) = Sum_{j=k..n} binomial(n,j)*Stirling_2(j,k)*Bell(n-j), where Bell(n) = A000110(n), for n >= 1, 0 <= k <= n-1. - N. J. A. Sloane, May 17 2016
a(2n,n) = A245109(n). - Alois P. Heinz, Aug 23 2017
Empirical: a(n,k) = p(1^n)[st(1^k)] (see A002872 for notation). - Andrey Zabolotskiy, Oct 17 2017
a(n,k) = Sum_{j=0..k} (-1)^(k-j)*A046716(k, k-j)*Bell(n + j)/k!. - Peter Luschny, Dec 06 2023

More terms from James Sellers.

Better definition from Geoffrey Critzer, Nov 30 2012.

A078937 Square of lower triangular matrix of A056857 (successive equalities in set partitions of n).

Original entry on oeis.org

1, 2, 1, 6, 4, 1, 22, 18, 6, 1, 94, 88, 36, 8, 1, 454, 470, 220, 60, 10, 1, 2430, 2724, 1410, 440, 90, 12, 1, 14214, 17010, 9534, 3290, 770, 126, 14, 1, 89918, 113712, 68040, 25424, 6580, 1232, 168, 16, 1, 610182, 809262, 511704, 204120, 57204, 11844, 1848, 216, 18, 1

Offset: 0

Paul D. Hanna, Dec 18 2002

First column gives A001861 (values of Bell polynomials); row sums gives A035009 (STIRLING transform of powers of 2);
Square of the matrix exp(P)/exp(1) given in A011971. - Gottfried Helms, Apr 08 2007. Base matrix in A011971 and in A056857, second power in this entry, third power in A078938, fourth power in A078939
Riordan array [exp(2*exp(x)-2),x], whose production matrix has e.g.f. exp(x*t)(t+2*exp(x)). [Paul Barry, Nov 26 2008]

			[0] 1;
[1] 2, 1;
[2] 6, 4, 1;
[3] 22, 18, 6, 1;
[4] 94, 88, 36, 8, 1;
[5] 454, 470, 220, 60, 10, 1;
[6] 2430, 2724, 1410, 440, 90, 12, 1;
[7] 14214, 17010, 9534, 3290, 770, 126, 14, 1;
[8] 89918, 113712, 68040, 25424, 6580, 1232, 168, 16, 1;

Cf. A056857, A001861, A035009.
Cf. A078938, A078944, A078945, A000110.
Cf. A078937, A078938, A129323, A129324, A129325, A027710.
Cf. A129327, A129328, A129329, A078944, A129331, A129332, A129333.

# Computes triangle as a matrix M(dim, p).
# A023531 (p=0), A056857 (p=1), this sequence (p=2), A078938 (p=3), ...
with(LinearAlgebra): M := (n, p) -> local j,k; MatrixPower(subs(exp(1) = 1,
MatrixExponential(MatrixExponential(Matrix(n, n, [seq(seq(`if`(j = k + 1, j, 0),
k = 0..n-1), j = 0..n-1)])))), p): M(8, 2);  # Peter Luschny, Mar 28 2024

k=9; m=matpascal(k)-matid(k+1); pe=matid(k+1)+sum(j=1,k,m^j/j!); A=pe^2; A /* Gottfried Helms, Apr 08 2007; amended by Georg Fischer Mar 28 2024 */

PE=exp(matpascal(5))/exp(1); A = PE^2; a(n)=A[n,column] with exact integer arithmetic: PE=exp(matpascal(5)-matid(6)); A = PE^2; a(n)=A[n,1] - Gottfried Helms, Apr 08 2007

Exponential function of 2*Pascal's triangle (taken as a lower triangular matrix) divided by e^2: [A078937] = (1/e^2)*exp(2*[A007318]) = [A056857]^2.

Entry revised by N. J. A. Sloane, Apr 25 2007

a(38) corrected by Georg Fischer, Mar 28 2024

A144180 Number of ways of placing n labeled balls into n unlabeled (but 5-colored) boxes.

Original entry on oeis.org

1, 5, 30, 205, 1555, 12880, 115155, 1101705, 11202680, 120415755, 1362057155, 16151603830, 200144023805, 2584429030505, 34691478901030, 483040313859705, 6963313750468055, 103747357497925880, 1595132080103893655

Offset: 0

Philippe Deléham, Sep 12 2008

nonn

a(n) is also the exp transform of A010716. - Alois P. Heinz, Oct 09 2008

The number of ways of putting n labeled balls into a set of bags and then putting the bags into 5 labeled boxes. - Peter Bala, Mar 23 2013

Alois P. Heinz, Table of n, a(n) for n = 0..200
H. D. Nguyen, D. Taggart, Mining the OEIS: Ten Experimental Conjectures, 2013; Mentions this sequence. - From _N. J. A. Sloane_, Mar 16 2014
N. J. A. Sloane, Transforms

Cf. A000110, A001861, A027710, A078944. A144223, A144263, A189233, A221159, A221176.

a:= proc(n) option remember; `if`(n=0, 1,
      (1+add(binomial(n-1, k-1)*a(n-k), k=1..n-1))*5)
    end:
seq(a(n), n=0..25); # Alois P. Heinz, Oct 09 2008

Table[BellB[n,5],{n,0,20}] (* Vaclav Kotesovec, Mar 12 2014 *)

expnums(19, 5) # Zerinvary Lajos, May 15 2009

a(n) = Sum_{k=0..n} 5^k * A048993(n,k); A048993: Stirling2 numbers.
G.f.: A(x) satisfies 5*(x/(1-x))*A(x/(1-x)) = A(x)-1; five times the binomial transform equals this sequence shifted one place left.
E.g.f.: exp(5*(exp(x)-1)).
G.f.: (G(0) - 1)/(x-1)/5 where G(k) =  1 - 5/(1-k*x)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 16 2013
a(n) ~ n^n * exp(n/LambertW(n/5)-5-n) / (sqrt(1+LambertW(n/5)) * LambertW(n/5)^n). - Vaclav Kotesovec, Mar 12 2014
G.f.: Sum_{j>=0} 5^j*x^j / Product_{k=1..j} (1 - k*x). - Ilya Gutkovskiy, Apr 07 2019

More terms from Alois P. Heinz, Oct 09 2008

A189233 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals upwards, where the e.g.f. of column k is exp(k*(e^x-1)).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 6, 3, 1, 0, 15, 22, 12, 4, 1, 0, 52, 94, 57, 20, 5, 1, 0, 203, 454, 309, 116, 30, 6, 1, 0, 877, 2430, 1866, 756, 205, 42, 7, 1, 0, 4140, 14214, 12351, 5428, 1555, 330, 56, 8, 1, 0, 21147, 89918, 88563, 42356, 12880, 2850, 497, 72, 9, 1

Offset: 0

Peter Luschny, Apr 18 2011

A(n,k) is the n-th moment of a Poisson distribution with mean = k. - Geoffrey Critzer, Dec 23 2018

			Square array begins:
       A000007 A000110 A001861 A027710 A078944 A144180 A144223 A144263
A000012   1,    1,    1,    1,    1,     1,     1,     1, ...
A001477   0,    1,    2,    3,    4,     5,     6,     7, ...
A002378   0,    2,    6,   12,   20,    30,    42,    56, ...
A033445   0,    5,   22,   57,  116,   205,   330,   497, ...
          0,   15,   94,  309,  756,  1555,  2850,  4809, ...
          0,   52,  454, 1866, 5428, 12880, 26682, 50134, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..5150
E. T. Bell, Exponential numbers, Amer. Math. Monthly, 41 (1934), 411-419.
Peter Luschny, Set partitions and Bell numbers

Columns: A000007, A000110, A001861, A027710, A078944, A144180, A144223, A144263.
Rows: A000012, A001477, A002378, A033445.
Main diagonal gives A242817.
Cf. A000587, A144150.

# Cf. also the Maple prog. of Alois P. Heinz in A144223 and A144180.
expnums := proc(k,n) option remember; local j;
`if`(n = 0, 1, (1+add(binomial(n-1,j-1)*expnums(k,n-j), j = 1..n-1))*k) end:
A189233_array := (k, n) -> expnums(k,n):
seq(print(seq(A189233_array(k,n), k = 0..7)), n = 0..5);
A189233_egf := k -> exp(k*(exp(x)-1));
T := (n,k) -> n!*coeff(series(A189233_egf(k), x, n+1), x, n):
seq(lprint(seq(T(n,k), k = 0..7)), n = 0..5):
# alternative Maple program:
A:= proc(n, k) option remember; `if`(n=0, 1,
      (1+add(binomial(n-1, j-1)*A(n-j, k), j=1..n-1))*k)
    end:
seq(seq(A(d-k, k), k=0..d), d=0..12);  # Alois P. Heinz, Sep 25 2017

max = 9; Clear[col]; col[k_] := col[k] = CoefficientList[ Series[ Exp[k*(Exp[x]-1)], {x, 0, max}], x]*Range[0, max]!; a[0, ] = 1; a[n?Positive, 0] = 0; a[n_, k_] := col[k][[n+1]]; Table[ a[n-k, k], {n, 0, max}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 26 2013 *)
Table[Table[BellB[n, k], {k, 0, 5}], {n, 0, 5}] // Grid  (* Geoffrey Critzer, Dec 23 2018 *)

A(n,k):=if k=0 and n=0 then 1 else if k=0 then 0 else  sum(stirling2(n,i)*k^i,i,0,n); /* Vladimir Kruchinin, Apr 12 2019 */

E.g.f. of column k: exp(k*(e^x-1)).
A(n,1) = A000110(n), A(n, -1) = A000587(n).
A(n,k) = BellPolynomial(n, k). - Geoffrey Critzer, Dec 23 2018
A(n,k) = Sum_{i=0..n} Stirling2(n,i)*k^i. - Vladimir Kruchinin, Apr 12 2019

A144223 Number of ways of placing n labeled balls into n unlabeled (but 6-colored) boxes.

Original entry on oeis.org

1, 6, 42, 330, 2850, 26682, 268098, 2869242, 32510850, 388109562, 4861622850, 63682081530, 869725707522, 12352785293562, 182049635623362, 2778394592545530, 43833623157604482, 713738052924821754

Offset: 0

Philippe Deléham, Sep 14 2008

nonn

a(n) is also the exp transform of A010722. - Alois P. Heinz, Oct 09 2008

The number of ways of putting n labeled balls into a set of bags and then putting the bags into 6 labeled boxes. - Peter Bala, Mar 23 2013

Alois P. Heinz, Table of n, a(n) for n = 0..200
N. J. A. Sloane, Transforms

Cf. A000110, A001861, A027710, A078944, A144180. A144263, A189233, A221159, A221176.

a:= proc(n) option remember; `if`(n=0, 1,
      (1+add(binomial(n-1, k-1)*a(n-k), k=1..n-1))*6)
    end:
seq(a(n), n=0..25); # Alois P. Heinz, Oct 09 2008

Table[BellB[n,6],{n,0,20}] (* Vaclav Kotesovec, Mar 12 2014 *)

expnums(18, 6) # Zerinvary Lajos, May 15 2009

a(n) = Sum_{k=0..n} 6^k*A048993(n,k); A048993: Stirling2 numbers.
G.f.: 6*(x/(1-x))*A(x/(1-x)) = A(x)-1; six times the binomial transform equals this sequence shifted one place left.
E.g.f.: exp(6(e^x-1)).
G.f.: T(0)/(1-6*x), where T(k) = 1 - 6*x^2*(k+1)/(6*x^2*(k+1) - (1-6*x-x*k)*(1-7*x-x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 04 2013
a(n) ~ n^n * exp(n/LambertW(n/6)-6-n) / (sqrt(1+LambertW(n/6)) * LambertW(n/6)^n). - Vaclav Kotesovec, Mar 12 2014
G.f.: Sum_{j>=0} 6^j*x^j / Product_{k=1..j} (1 - k*x). - Ilya Gutkovskiy, Apr 07 2019

More terms from Alois P. Heinz, Oct 09 2008

A144263 Number of ways of placing n labeled balls into n unlabeled (but7-colored) boxes.

Original entry on oeis.org

1, 7, 56, 497, 4809, 50134, 558215, 6593839, 82187658, 1076193867, 14749823893, 210926792244, 3138696242941, 48485723853763, 775929767223352, 12840232627455485, 219355194338036309, 3862794707291567670

Offset: 0

Philippe Deléham, Sep 16 2008

nonn

a(n) is also the exp transform of A010727. - Alois P. Heinz, Oct 09 2008

The number of ways of putting n labeled balls into a set of bags and then putting the bags into 7 labeled boxes. - Peter Bala, Mar 23 2013

Alois P. Heinz, Table of n, a(n) for n = 0..200
N. J. A. Sloane, Transforms

Cf. A000110, A001861, A027710, A078944, A144180, A144223. A189233, A221159, A221176.

a:= proc(n) option remember; `if`(n=0, 1,
      (1+add(binomial(n-1, k-1)*a(n-k), k=1..n-1))*7)
    end:
seq(a(n), n=0..25); # Alois P. Heinz, Oct 09 2008

Table[BellB[n,7],{n,0,20}] (* Vaclav Kotesovec, Mar 12 2014 *)

expnums(18, 7) # Zerinvary Lajos, May 15 2009

a(n) = Sum_{k=0..n} 7^k*A048993(n,k); A048993: Stirling2 numbers.
E.g.f.: exp(7*(exp(x)-1)).
G.f.: 7*(x/(1-x))*A(x/(1-x))= A(x)-1; seven times the binomial transform equals this sequence shifted one place left.
a(n) ~ n^n * exp(n/LambertW(n/7)-7-n) / (sqrt(1+LambertW(n/7)) * LambertW(n/7)^n). - Vaclav Kotesovec, Mar 12 2014
G.f.: Sum_{j>=0} 7^j*x^j / Product_{k=1..j} (1 - k*x). - Ilya Gutkovskiy, Apr 11 2019

More terms from Alois P. Heinz, Oct 09 2008

cp's OEIS Frontend

A001883 Number of permutations s of {1,2,...,n} such that |s(i)-i|>1 for each i=1,2,...,n.

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A005494 3-Bell numbers: E.g.f.: exp(3*z + exp(z) - 1).

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A109128 Triangle read by rows: T(n,k) = T(n-1,k-1) + T(n-1,k) + 1 for 0

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A035009 STIRLING transform of [1,1,2,4,8,16,32,...].

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A049020 Triangle of numbers a(n,k), 0 <= k <= n: number of set partitions of {1,2,...,n} in which exactly k of the blocks have been distinguished.

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A078937 Square of lower triangular matrix of A056857 (successive equalities in set partitions of n).

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