A020556 -id:A020556 - cp's OEIS frontend

A071379 a(n) = (1/e) * Sum_{k>=0} ((k+4)!/k!)^(n-1)/k!.

1, 1, 209, 163121, 326922081, 1346634725665, 9939316337679281, 119802044788535500753, 2205421644124274191535553, 58945667435045762187763602753, 2198513228897522394476415669503377, 110833342180980170285766876408530089329

Offset: 0

Karol A. Penson, May 22 2002

nonn

This is a Dobinski-type summation formula.
Terms quickly become gigantic: a(15) = 9142140479823239889945170786704021785456107245847570873873. a(n) appears in the process of ordering the n-th power of a product of fourth power of boson creation and fourth power of boson annihilation operators.
From Peter Luschny, Mar 27 2011: (Start)
Let B_{m}(x) = Sum_{j>=0} (exp(j!/(j-m)!*x-1)/j!) then a(n) = n! [x^n] Taylor(B_{4}(x)), where [x^n] denotes the coefficient of x^k in the Taylor series for B_{4}(x).
a(n) is row 4 of the square array representation of A090210. (End)

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
K. A. Penson, P. Blasiak, A. Horzela, A. I. Solomon and G. H. E. Duchamp, Laguerre-type derivatives: Dobinski relations and combinatorial identities, J. Math. Phys. 50, 083512 (2009).
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Cf. A000110, A020556 and A069223, when k+4 is replaced by k+1, k+2 or k+3 respectively.

Cf. A090210.

A071379 := proc(n) local r,s,i;
if n=0 then 1 else r := [seq(5,i=1..n-1)]; s := [seq(1,i=1..n-1)];
exp(-x)*24^(n-1)*hypergeom(r,s,x); round(evalf(subs(x=1,%),99)) fi end:
seq(A071379(n),n=0..10); # Peter Luschny, Mar 30 2011

a[n_] := Sum[FactorialPower[k, 4]^n/k!, {k, 4, Infinity}]/E; a[0] = 1; Array[a, 12, 0] (* Jean-François Alcover, Sep 01 2016 *)

default(realprecision, 500); for(n=0, 20, print1(if(n==0, 1, round(exp(-1)*sum(k=0, 500, ((k+4)!/k!)^(n-1)/k!))), ", ")) \\ G. C. Greubel, May 15 2018

a(n) = (1/e)*Sum_{k>=4} fallfac(k, 4)^n / k!, n >= 1, with fallfac(n, m) := A008279(n, m) (falling factorials). (From eq.(26) with r=4 of the Schork reference.)

E.g.f. with a(0) := 1: (1/e)*(Sum_{k>=4} e^(fallfac(k, 4)*x)/k! + 8/3). From top of p. 4656 with r=4 of the Schork reference.

a(0)=1 prepended by Alois P. Heinz, Aug 01 2016

If it is proved that A283153 and A071379 are the same, then the entries should be merged and A283153 recycled. - N. J. A. Sloane, Mar 06 2017

A208021 T(n,k) = Number of n X k nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal, vertical, diagonal or antidiagonal neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 5, 7, 7, 5, 15, 87, 270, 87, 15, 52, 1657, 27093, 27093, 1657, 52, 203, 43833, 5252041, 30066912, 5252041, 43833, 203, 877, 1515903, 1688298227, 80318704605, 80318704605, 1688298227, 1515903, 877, 4140, 65766991

Offset: 1

R. H. Hardin, Feb 22 2012

Equivalently, the number of colorings of the n x k king graph using any number of colors up to permutation of the colors. - Andrew Howroyd, Jun 25 2017

			Table starts
...1........1............2...............5...............15..............52
...1........1............7..............87.............1657...........43833
...2........7..........270...........27093..........5252041......1688298227
...5.......87........27093........30066912......80318704605.421673189900658
..15.....1657......5252041.....80318704605.3662498214110836
..52....43833...1688298227.421673189900658
.203..1515903.819147302097
.877.65766991
...
Some solutions for n=4 k=3
..0..1..2....0..1..0....0..1..0....0..1..0....0..1..0....0..1..0....0..1..2
..2..3..4....2..3..2....2..3..2....2..3..2....2..3..2....2..3..2....2..3..0
..5..6..0....4..0..4....0..1..0....4..1..0....0..1..0....0..4..0....4..5..4
..2..3..1....1..2..1....2..3..4....5..2..3....2..4..2....1..2..1....0..1..2

Andrew Howroyd, Table of n, a(n) for n = 1..231 (terms 1..49 from R. H. Hardin)
Eric Weisstein's World of Mathematics, King Graph

Columns 1-5 are A000110(n-1), A020556(n-1), A208018, A208019, A208020.
Main diagonal is A289136.
Cf. A207868, A212208, A212209.

A216078 Number of horizontal and antidiagonal neighbor colorings of the odd squares of an n X 2 array with new integer colors introduced in row major order.

Original entry on oeis.org

1, 1, 3, 7, 27, 87, 409, 1657, 9089, 43833, 272947, 1515903, 10515147, 65766991, 501178937, 3473600465, 28773452321, 218310229201, 1949230218691, 16035686850327, 153281759047387, 1356791248984295, 13806215066685433, 130660110400259849, 1408621900803060705

Offset: 1

R. H. Hardin, Sep 01 2012

nonn

Number of vertex covers and independent vertex sets of the n-1 X n-1 white bishops graph. Equivalently, the number of ways to place any number of non-attacking bishops on the white squares of an n-1 X n-1 board. - Andrew Howroyd, May 08 2017

Number of pairs of partitions (A<=B) of [n-1] such that the nontrivial blocks of A are of type {k,n-1-k} if n is even, and of type {k,n-k} if n is odd. - Francesca Aicardi, May 28 2022

			Some solutions for n = 5:
  x 0   x 0   x 0   x 0   x 0   x 0   x 0   x 0   x 0   x 0
  1 x   1 x   1 x   1 x   1 x   1 x   1 x   1 x   1 x   1 x
  x 2   x 0   x 0   x 2   x 0   x 1   x 1   x 2   x 2   x 1
  0 x   2 x   1 x   3 x   1 x   0 x   2 x   3 x   0 x   0 x
  x 3   x 1   x 2   x 2   x 0   x 1   x 1   x 1   x 2   x 0
There are 4 white squares on a 3 X 3 board. There is 1 way to place no non-attacking bishops, 4 ways to place 1 and 2 ways to place 2 so a(4) = 1 + 4 + 2 = 7. - _Andrew Howroyd_, Jun 06 2017

R. H. Hardin, Table of n, a(n) for n = 1..210
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Vertex Cover
Eric Weisstein's World of Mathematics, White Bishop Graph

Column 2 of A216084.
Row sums of A274106(n-1).
Cf. A020556, A094577, A216332, A201862, A286423.

a:= n-> (m-> add(binomial(m, k)*combinat[bell](m+k+e)
           , k=0..m))(iquo(n-1, 2, 'e')):
seq(a(n), n=1..26);  # Alois P. Heinz, Oct 03 2022

a[n_] := Module[{m, e}, {m, e} = QuotientRemainder[n - 1, 2];
   Sum[Binomial[m, k]*BellB[m + k + e], {k, 0, m}]];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Jul 25 2022, after Francesca Aicardi *)

a(n) = Sum_{k=0..m} binomial(m, k)*Bell(m+k+e), with m = floor((n-1)/2), e = (n+1) mod 2 and where Bell(n) is the Bell exponential number A000110(n). - Francesca Aicardi, May 28 2022
From Vaclav Kotesovec, Jul 29 2022: (Start)
a(2*k) = A020556(k).
a(2*k+1) = A094577(k). (End)

A069948 a(n) = 1/exp(1) * Sum_{k>=0} (k+n)!^2 / k!^3.

Original entry on oeis.org

1, 5, 87, 2971, 163121, 12962661, 1395857215, 194634226067, 33990369362241, 7247035915622821, 1848636684656077991, 555005864462114884875, 193458213840943964983537, 77399534126148191747554181, 35196002960227350045891984591, 18037244723394790042393195636291

Offset: 0

Robert G. Wilson v, May 02 2002

nonn

From Peter Luschny, Mar 27 2011: (Start)
Let B_{n}(x) = sum_{j>=0}(exp(j!/(j-n)!*x-1)/j!) then a(n) = 3! [x^3] taylor(B_{n}(x)), where [x^3] denotes the coefficient of x^3 in the Taylor series for B_{n}(x).
a(n) is column 3 of the square array representation of A090210. (End)

G. C. Greubel, Table of n, a(n) for n = 0..240
K. A. Penson, P. Blasiak, A. Horzela, G. H. E. Duchamp and A. I. Solomon, Laguerre-type derivatives: Dobinski relations and combinatorial identities, J. Math. Phys. vol. 50, 083512 (2009)

Cf. A000110, A020556, A069223, A090210.

A069948 := proc(n) exp(-x)*n!^2*hypergeom([n+1,n+1],[1,1],x);
round(evalf(subs(x=1,%),99)) end:
seq(A069948(n),n=0..13); # Peter Luschny, Mar 30 2011
# second Maple program:
a:= n-> sum((k+n)!^2/k!^3, k=0..infinity)/exp(1):
seq(a(n), n=0..15);  # Alois P. Heinz, May 17 2018

f[n_] := f[n] = Sum[(k + n)!^3/((k + n)!*(k!^3)*E), {k, 0, Infinity}]; Table[ f[n], {n, 0, 13}] (* or *)
Table[n!^2*HypergeometricPFQ[{n + 1, n + 1}, {1, 1}, 1]/Exp[1], {n, 0, 13}] (* Robert G. Wilson v, Jan 11 2007 *)

{default(realprecision, 200)}; for(n=0,30, print1(round(exp(-1)*(n!)^2*sum(k=0,500, binomial(n+k, k)^2/k!)), ", ")) \\ G. C. Greubel, May 17 2018

Integral representation as n-th moment of a positive function on a positive halfaxis (solution of the Stieltjes moment problem), in Maple notation: a(n)=int(x^n*2*BesselK(0,2*sqrt(x))*hypergeom([],[1,1],x)/exp(1), x=0..infinity), n=0,1... Special values of the hypergeometric function of type 2F2: a(n)=exp(-1)*GAMMA(n+1)^2*hypergeom([n+1, n+1], [1, 1], 1). - Karol A. Penson and G. H. E. Duchamp (gduchamp2(AT)free.fr), Jan 09 2007
Recurrence: (8*n-7)*a(n) = (24*n^3 + 3*n^2 - 26*n + 4)*a(n-1) - (n-1)^2*(24*n^3 - 85*n^2 + 66*n + 13)*a(n-2) + (n-1)^2*(8*n+1)*(n-2)^4*a(n-3). - Vaclav Kotesovec, Jul 30 2013
a(n) ~ n^(2*n+1/3)*exp(n^(1/3) + 3*n^(2/3) - 2*n - 2/3)/sqrt(3) * (1 + 41/(54*n^(1/3)) + 13769/(29160*n^(2/3))). - Vaclav Kotesovec, Jul 30 2013

More terms from Robert G. Wilson v, Jan 11 2007

A090209 Generalized Bell numbers (from (5,5)-Stirling2 array A090216).

Original entry on oeis.org

1, 1, 1546, 12962661, 363303011071, 25571928251231076, 3789505947767235111051, 1049433111253356296672432821, 498382374325731085522315594481036, 380385281554629647028734545622539438171, 443499171330317702437047276255605780991365151

Offset: 0

Wolfdieter Lang, Dec 01 2003

Contribution from Peter Luschny, Mar 27 2011: (Start) Let B_{m}(x) = sum_{j>=0}(exp(j!/(j-m)!*x-1)/j!) then a(n) = n! [x^n] taylor(B_{5}(x)), where [x^n] denotes the coefficient of x^k in the Taylor series for B_{5}(x).

a(n) is row 5 of the square array representation of A090210. (End)

M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

G. C. Greubel, Table of n, a(n) for n = 0..115
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem., arXiv:quant-ph/0402027, Phys. Lett. A 309 (3-4) (2003) 198-205
K. A. Penson, P. Blasiak, A. Horzela, A. I. Solomon and G. H. E. Duchamp, Laguerre-type derivatives: Dobinski relations and combinatorial identities, J. Math. Phys. 50, 083512 (2009).

Cf. (Generalized) Bell numbers from (m,m)-Stirling2 array: A000110 (m=1), A020556 (m=2), A069223 (m=3), A071379 (m=4). Triangle A090210.

A071379 := proc(n) local r,s,i;
if n=0 then 1 else r := [seq(6,i=1..n-1)]; s := [seq(1,i=1..n-1)];
exp(-x)*5!^(n-1)*hypergeom(r,s,x); round(evalf(subs(x=1,%),99)) fi end:
seq(A071379(n),n=0..8); # Peter Luschny, Mar 30 2011

fallfac[n_, k_] := Pochhammer[n-k+1, k]; a[n_, k_] := (((-1)^k)/k!)*Sum[((-1)^p)*Binomial[k, p]*fallfac[p, 5]^n, {p, 5, k}]; a[0] = 1; a[n_] := Sum[a[n, k], {k, 5, 5*n}];  Table[a[n], {n, 0, 8}] (* Jean-François Alcover, Mar 05 2014 *)

a(n) = Sum_{k=5..5*n} A090216(n, k), n>=1. a(0) := 1.
a(n) = Sum_{k >=5} (fallfac(k, 5)^n)/k!/exp(1), n>=1, a(0) := 1. From eq.(26) with r=5 of the Schork reference.
E.g.f. with a(0) := 1: (sum((exp(fallfac(k, 5)*x))/k!, k=5..infinity)+ A000522(4)/4!)/exp(1). From the top of p. 4656 with r=5 of the Schork reference.

If it is proved that A283154 and A090209 are the same, then the entries should be merged and A283154 recycled. - N. J. A. Sloane, Mar 06 2017

A182933 Generalized Bell numbers based on the rising factorial powers; square array read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 5, 27, 13, 1, 1, 15, 409, 778, 73, 1, 1, 52, 9089, 104149, 37553, 501, 1, 1, 203, 272947, 25053583, 57184313, 2688546, 4051, 1, 1, 877, 10515147, 9566642254, 192052025697, 56410245661, 265141267, 37633, 1

Offset: 0

Peter Luschny, Mar 29 2011

These numbers are related to the generalized Bell numbers based on the falling factorial powers (A090210).
The square array starts for n>=0, k>=0:
n\k=0,1,..  A000012,A000262,A182934,...
A000012: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
A000110: 1, 1, 2, 5, 15, 52, 203, 877, 4140, ...
A094577: 1, 3, 27, 409, 9089, 272947, 10515147, ...
A182932: 1, 13, 778, 104149, 25053583, 9566642254, ...
       : 1, 73, 37553, 57184313, 192052025697, ...
       : 1, 501, 2688546, 56410245661, ...
 ....  : 1, 4051, 265141267, 89501806774945, ...

Cf. A000110, A020556, A069223, A071379, A090209, A002720, A069948, A182925, A182924, A182933.

A182933_AsSquareArray := proc(n,k) local r,s,i;
r := [seq(n+1,i=1..k)]; s := [seq(1,i=1..k-1),2];
exp(-x)*n!^k*hypergeom(r,s,x); round(evalf(subs(x=1,%),99)) end:
seq(lprint(seq(A182933_AsSquareArray(n,k),k=0..6)),n=0..6);

a[n_, k_] := Exp[-1]*n!^k*HypergeometricPFQ[ Table[n+1, {k}], Append[ Table[1, {k-1}], 2], 1.]; Table[ a[n-k, k] // Round , {n, 0, 8}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)

Let r_k = [n+1,...,n+1] (k occurrences of n+1), s_k = [1,...,1,2] (k-1 occurrences of 1) and F_k the generalized hypergeometric function of type k_F_k, then a(n,k) = exp(-1)*n!^k*F_k(r_k, s_k | 1).

Let B_{n}(x) = sum_{j>=0}(exp((j+n-1)!/(j-1)!*x-1)/j!) then a(n,k) = k! [x^k] series(B_{n}(x)), where [x^k] denotes the coefficient of x^k in the Taylor series for B_{n}(x).

A283153 Number of set partitions of unique elements from an n X 4 matrix where elements from the same row may not be in the same partition.

Original entry on oeis.org

1, 209, 163121, 326922081, 1346634725665, 9939316337679281, 119802044788535500753, 2205421644124274191535553, 58945667435045762187763602753, 2198513228897522394476415669503377, 110833342180980170285766876408530089329, 7356710448423295420590529054176924329802529, 628972339934967292421997567343442748145219556449

Offset: 1

Marko Riedel, Mar 01 2017

nonn

Apparently a duplicate of A071379? - R. J. Mathar, Mar 06 2017

Cf. A283154, A283155, A071379.

Table[(4!^n) * Sum[Binomial[p,4]^n/p! * Sum[(-1)^k/k!,{k,0,4n-p}],{p,1,4n}],{n,1,50}] (* Indranil Ghosh, Mar 04 2017 *)

a(n) = (4!^n) * sum(p=1, 4*n, binomial(p,4)^n/p! * sum(k=0, 4*n-p, (-1)^k/k!)); \\ Indranil Ghosh, Mar 04 2017

a(n) = m!^n * Sum_{p=1..n*m} (binomial(p,m)^n/p!) * Sum_{k=0..n*m-p} (-1)^k/k! with m=4.

If it is proved that A283153 and A071379 are the same, then the entries should be merged and A283153 recycled. - N. J. A. Sloane, Mar 06 2017

A090211 Alternating row sums of array A078739 ((2,2)-Stirling2).

Original entry on oeis.org

1, -1, -1, 41, -375, -3001, 177063, -990543, -144800527, 3644593711, 214013895023, -12488200175463, -553322483517383, 61495192102867639, 2469939623420627543, -448608666325921194271, -19104207797417792353951, 4742067751530355028847327

Offset: 1

Wolfdieter Lang, Dec 01 2003

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem.

Cf. -A000587(n) from Stirling2 case A008277 with a(0) := -1. A020556 (non-alternating sum, generalized Bell-numbers).

a[n_] := Sum[(-1)^k FactorialPower[k, 2]^n/k!, {k, 2, Infinity}]*E; Array[a, 18] (* Jean-François Alcover, Sep 01 2016 *)

a(n) := sum( A078739(n, m)*(-1)^m, m=2..2*n), n>=1. a(0) := +1 may be added.
a(n) = sum(((-1)^k)*(fallfac(k, 2)^n)/k!, k=2..infinity)*exp(1), with fallfac(k, 2)=A008279(k, 2)=k*(k-1) and n>=1. This produces also a(0)=1.
E.g.f. if a(0)=1 is added: exp(1)*(sum(((-1)^k)*exp(k*(k-1)*x)/k!, k=2..infinity)). Similar to derivation on top p. 4656 of the Schork reference.

A098631 Consider the family of directed multigraphs enriched by the species of parts. Sequence gives number of those multigraphs with n labeled arcs.

Original entry on oeis.org

1, 2, 28, 696, 26512, 1402656, 97017792, 8418174848, 889241719040, 111774837350912, 16420543334734848, 2778708477919836160, 535183812199464341504, 116142946557502449852416, 28156854547845767203373056, 7569375509914847295271043072, 2241898693518356603925445017600

Offset: 0

N. J. A. Sloane, Oct 26 2004

nonn

G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.

Andrew Howroyd, Table of n, a(n) for n = 0..100
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]

Cf. A000079, A014505, A020556, A098623, A098628, A098629, A098630.

\\ R(n) is A000079 as e.g.f.; EnrichedGdSeq defined in A098623.
R(n)={exp(2*x + O(x*x^n))}
EnrichedGdSeq(R(20)) \\ Andrew Howroyd, Jan 12 2021

a(n) = 2^n*A020556(n). - Vladeta Jovovic, Aug 11 2005

E.g.f.: B(R(x)) where B(x) is the e.g.f. of A014505 and 1 + R(x) is the e.g.f. of A000079. - Andrew Howroyd, Jan 12 2021

More terms from Vladeta Jovovic, Aug 11 2005

Terms a(14) and beyond from Andrew Howroyd, Jan 12 2021

A182930 Triangle read by rows: Number of set partitions of {1,2,..,n} such that |k| is a block and no block |m| with m < k exists, (1 <= n, 1 <= k <= n).

Original entry on oeis.org

1, 1, 0, 2, 1, 1, 5, 3, 2, 1, 15, 10, 7, 5, 4, 52, 37, 27, 20, 15, 11, 203, 151, 114, 87, 67, 52, 41, 877, 674, 523, 409, 322, 255, 203, 162, 4140, 3263, 2589, 2066, 1657, 1335, 1080, 877, 715, 21147, 17007, 13744, 11155, 9089, 7432, 6097, 5017, 4140, 3425

Offset: 1

Peter Luschny, Apr 08 2011

Mirror image of A106436. - Alois P. Heinz, Jan 29 2019

			T(4,2) = card({2|134, 2|3|14, 2|4|13}) = 3.
[1]     1,
[2]     1,    0,
[3]     2,    1,    1,
[4]     5,    3,    2,    1,
[5]    15,   10,    7,    5,    4,
[6]    52,   37,   27,   20,   15,   11,
     [-1-] [-2-] [-3-] [-4-] [-5-] [-6-]

Alois P. Heinz, Rows n = 1..141, flattened
Peter Luschny, Set partitions

Cf. A000110, A000296, A106436.

T(2n+1,n+1) gives A020556.

T := proc(n, k) option remember; if n = 1 then 1 elif n = k then T(n-1,1) - T(n-1,n-1) else T(n-1,k) + T(n, k+1) fi end:
A182930 := (n,k) -> T(n,k); seq(print(seq(A182930(n,k),k=1..n)),n=1..6);

T[n_, k_] := T[n, k] = Which[n == 1, 1, n == k, T[n-1, 1] - T[n-1, n-1], True, T[n-1, k] + T[n, k+1]];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] (* Jean-François Alcover, Jun 22 2019 *)

Recursion: The value of T(n,k) is, if n < 0 or k < 0 or k > n undefined, else if n = 1 then 1 else if k = n then T(n-1,1) - T(n-1,n-1); in all other cases T(n,k) = T(n,k+1) + T(n-1,k).

cp's OEIS Frontend

A071379 a(n) = (1/e) * Sum_{k>=0} ((k+4)!/k!)^(n-1)/k!.

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A208021 T(n,k) = Number of n X k nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal, vertical, diagonal or antidiagonal neighbor (colorings ignoring permutations of colors).

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A216078 Number of horizontal and antidiagonal neighbor colorings of the odd squares of an n X 2 array with new integer colors introduced in row major order.

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A069948 a(n) = 1/exp(1) * Sum_{k>=0} (k+n)!^2 / k!^3.

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A090209 Generalized Bell numbers (from (5,5)-Stirling2 array A090216).

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A182933 Generalized Bell numbers based on the rising factorial powers; square array read by antidiagonals.

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A283153 Number of set partitions of unique elements from an n X 4 matrix where elements from the same row may not be in the same partition.

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