A271466 -id:A271466 - cp's OEIS frontend

A186021 a(n) = Bell(n)*(2 - 0^n).

1, 2, 4, 10, 30, 104, 406, 1754, 8280, 42294, 231950, 1357140, 8427194, 55288874, 381798644, 2765917090, 20960284294, 165729739608, 1364153612318, 11665484410114, 103448316470744, 949739632313502, 9013431476894646, 88304011710168692, 891917738589610578, 9277180664459998706

Offset: 0

Paul Barry, Feb 10 2011

a(n) is the number of collections of subsets of {1,2,...,n-1} that are pairwise disjoint. a(n+1) = 2*Sum_{j=0..n} C(n,j)*Bell(j). For example a(3)=10 because we have: {}, {{}}, {{1}}, {{2}}, {{1,2}}, {{},{1}}, {{},{2}}, {{},{1,2}}, {{1},{2}}, {{},{1},{2}}. - Geoffrey Critzer, Aug 28 2014

a(n) is the number of collections of subsets of [n] that are pairwise disjoint and cover [n], with [0] = {}. For disjoint collections of nonempty subsets see A000110. For arbitrary collections of subsets see A000371. For arbitrary collections of nonempty subsets see A003465. - Manfred Boergens, May 02 2024 and Apr 09 2025

			a(4) = A060719(3) + 1 = 29 + 1 = 30.

G. C. Greubel, Table of n, a(n) for n = 0..575

Row sums of A186020 and A256894.
Main diagonal of A271466 (shifted) and A381682.
Cf. A000110, A000371, A003465.

[Bell(n)*(2-0^n): n in [0..50]]; // Vincenzo Librandi, Apr 06 2011

A186021List := proc(m) local A, P, n; A := [1,2]; P := [2];
for n from 1 to m - 2 do P := ListTools:-PartialSums([P[-1], op(P)]);
A := [op(A), P[-1]] od; A end: A186021List(26); # Peter Luschny, Mar 24 2022

Prepend[Table[2 Sum[Binomial[n, j] BellB[j], {j, 0, n}], {n, 0, 25}], 1] (* Geoffrey Critzer, Aug 28 2014 *)
With[{nmax = 50}, CoefficientList[Series[2*Exp[Exp[x] - 1] - 1, {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Jul 24 2017 *)

x='x+O('x^50); Vec(serlaplace(2*exp(exp(x) - 1) -1)) \\ G. C. Greubel, Jul 24 2017

from itertools import accumulate
def A186021_list(size):
    if size < 1: return []
    L, accu = [1], [2]
    for _ in range(size-1):
        accu = list(accumulate([accu[-1]] + accu))
        L.append(accu[0])
    return L
print(A186021_list(26)) # Peter Luschny, Apr 25 2016

E.g.f.: 2*exp(exp(x)-1)-1. - Paul Barry, Apr 06 2011
a(n) = A000110(n)*A040000(n).
a(n+1) = 1 + Sum_{k=0..n} C(n,k)*a(k). - Franklin T. Adams-Watters, Oct 02 2011
From Sergei N. Gladkovskii, Nov 11 2012 to Mar 29 2013: (Start)
Continued fractions:
G.f.: A(x)= 1 + 2*x/(G(0)-x) where G(k)= 1 - x*(k+1)/(1 - x/G(k+1)).
G.f.: G(0)-1 where G(k) = 1-(x*k+1)/(x*k - 1 - x*(x*k - 1)/(x + (x*k + 1)/G(k+1))).
G.f.: (G(0)-2)/x - 1 where G(k) = 1 + 1/(1 - x/(x + (1 - x*k)/G(k+1))).
G.f.: (S-2)/x - 1 where S = 2*Sum_{k>=0} x^k/Product_{n=0..k-1}(1 - n*x).
G.f.: 1/(1-x) - x/(G(0)-x^2+x) where G(k) =x^2 + x - 1 + k*(2*x-x^2) - x^2*k^2 + x*(x*k - 1)*(x*k + 2*x - 1)^2/G(k+1).
E.g.f.: E(0) - 1 where E(k) = 1 + 1/(1 - 1/(1 + (k+1)/x*Bell(k)/Bell(k+1)/E(k+1))). (End)
a(n) = A060719(n-1) + 1, and the inverse binomial transform of A060719. - Gary W. Adamson, May 20 2013
G.f. A(x) satisfies: A(x) = 1 + (x/(1 - x)) * (1 + A(x/(1 - x))). - Ilya Gutkovskiy, Jun 30 2020

A095149 Triangle read by rows: Aitken's array (A011971) but with a leading diagonal before it given by the Bell numbers (A000110), 1, 1, 2, 5, 15, 52, ...

Original entry on oeis.org

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

Offset: 0

Gary W. Adamson, May 30 2004

Or, prefix Aitken's array (A011971) with a leading diagonal of 0's and take the differences of each row to get the new triangle.
With offset 1, triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} in which k is the largest entry in the block containing 1 (1 <= k <= n). - Emeric Deutsch, Oct 29 2006
Row term sums = the Bell numbers starting with A000110(1): 1, 2, 5, 15, ...
The k-th term in the n-th row is the number of permutations of length n starting with k and avoiding the dashed pattern 23-1. Equivalently, the number of permutations of length n ending with k and avoiding 1-32. - Andrew Baxter, Jun 13 2011
From Gus Wiseman, Aug 11 2020: (Start)
Conjecture: Also the number of divisors d with distinct prime multiplicities of the superprimorial A006939(n) that are of the form d = m * 2^k where m is odd. For example, row n = 4 counts the following divisors:
  1     2     4    8     16
  3     18    12   24    48
  5     50    20   40    80
  7     54    28   56    112
  9     1350  108  72    144
  25          540  200   400
  27          756  360   432
  45               504   720
  63               600   1008
  75               1400  1200
  135                    2160
  175                    2800
  189                    3024
  675                    10800
  4725                   75600
Equivalently, T(n,k) is the number of length-n vectors 0 <= v_i <= i whose nonzero values are distinct and such that v_n = k.
Crossrefs:
A008278 is the version counted by omega A001221.
A336420 is the version counted by Omega A001222.
A006939 lists superprimorials or Chernoff numbers.
A008302 counts divisors of superprimorials by Omega.
A022915 counts permutations of prime indices of superprimorials.
A098859 counts partitions with distinct multiplicities.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
Cf. A000005, A000142, A027423, A076954, A124010, A146291, A181818, A336417, A336419, A336421, A336499, A336942.
(End)

			Triangle starts:
   1;
   1,  1;
   2,  1,  2;
   5,  2,  3,  5;
  15,  5,  7, 10, 15;
  52, 15, 20, 27, 37, 52;
From _Gus Wiseman_, Aug 11 2020: (Start)
Row n = 3 counts the following set partitions (described in Emeric Deutsch's comment above):
  {1}{234}      {12}{34}    {123}{4}    {1234}
  {1}{2}{34}    {12}{3}{4}  {13}{24}    {124}{3}
  {1}{23}{4}                {13}{2}{4}  {134}{2}
  {1}{24}{3}                            {14}{23}
  {1}{2}{3}{4}                          {14}{2}{3}
(End)

Alois P. Heinz, Rows n = 0..150, flattened (first 51 rows from Chai Wah Wu)
Andrew M. Baxter and Lara K. Pudwell, Enumeration schemes for dashed patterns, arXiv:1108.2642 [math.CO], 2011.
Anders Claesson, Generalized pattern avoidance, Europ. J. Combin., 22 7 (2001), 961-971. See Proposition 3.
A. Bernini, M. Bouvel and L. Ferrari, Some statistics on permutations avoiding generalized patterns, PU.M.A. Vol. 18 (2007), No. 3-4, pp. 223-237 (see transposed array p. 227).

Cf. A000110, A005493, A008278, A011971, A188919, A271466.

T(2n,n) gives A020556.

with(combinat): T:=proc(n,k) if k=1 then bell(n-1) elif k=2 and n>=2 then bell(n-2) elif k<=n then add(binomial(k-2,i)*bell(n-2-i),i=0..k-2) else 0 fi end: matrix(8,8,T): for n from 1 to 11 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
Q[1]:=t*s: for n from 2 to 11 do Q[n]:=expand(t^n*subs(t=1,Q[n-1])+s*diff(Q[n-1],s)-Q[n-1]+s*Q[n-1]) od: for n from 1 to 11 do P[n]:=sort(subs(s=1,Q[n])) od: for n from 1 to 11 do seq(coeff(P[n],t,k),k=1..n) od; # yields sequence in triangular form - Emeric Deutsch, Oct 29 2006
A011971 := proc(n,k) option remember ; if k = 0 then if n=0 then 1; else A011971(n-1,n-1) ; fi ; else A011971(n,k-1)+A011971(n-1,k-1) ; fi ; end: A000110 := proc(n) option remember; if n<=1 then 1 ; else add( binomial(n-1,i)*A000110(n-1-i),i=0..n-1) ; fi ; end: A095149 := proc(n,k) option remember ; if k = 0 then A000110(n) ; else A011971(n-1,k-1) ; fi ; end: for n from 0 to 11 do for k from 0 to n do printf("%d, ",A095149(n,k)) ; od ; od ; # R. J. Mathar, Feb 05 2007
# alternative Maple program:
b:= proc(n, m, k) option remember; `if`(n=0, 1, add(
      b(n-1, max(j, m), max(k-1, -1)), j=`if`(k=0, m+1, 1..m+1)))
    end:
T:= (n, k)-> b(n, 0, n-k):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Dec 20 2018

nmax = 10; t[n_, 1] = t[n_, n_] = BellB[n-1]; t[n_, 2] = BellB[n-2]; t[n_, k_] /; n >= k >= 3 := t[n, k] = t[n, k-1] + t[n-1, k-1]; Flatten[ Table[ t[n, k], {n, 1, nmax}, {k, 1, n}]] (* Jean-François Alcover, Nov 15 2011, from formula with offset 1 *)

# requires Python 3.2 or higher.
from itertools import accumulate
A095149_list, blist = [1,1,1], [1]
for _ in range(2*10**2):
    b = blist[-1]
    blist = list(accumulate([b]+blist))
    A095149_list += [blist[-1]]+ blist
# Chai Wah Wu, Sep 02 2014, updated Chai Wah Wu, Sep 20 2014

With offset 1, T(n,1) = T(n,n) = T(n+1,2) = B(n-1) = A000110(n-1) (the Bell numbers). T(n,k) = T(n,k-1) + T(n-1,k-1) for n >= k >= 3. T(n,n-1) = B(n-1) - B(n-2) = A005493(n-3) for n >= 3 (B(q) are the Bell numbers A000110). T(n,k) = A011971(n-2,k-2) for n >= k >= 2. In other words, deleting the first row and first column we obtain Aitken's array A011971 (also called Bell or Pierce triangle; offset in A011971 is 0). - Emeric Deutsch, Oct 29 2006

T(n,1) = B(n-1); T(n,2) = B(n-2) for n >= 2; T(n,k) = Sum_{i=0..k-2} binomial(k-2,i)*B(n-2-i) for 3 <= k <= n, where B(q) are the Bell numbers (A000110). Generating polynomial of row n is P[n](t) = Q[n](t,1), where Q[n](t,s) = t^n*Q[n-1](1,s) + s*dQ[n-1](t,s)/ds + (s-1) Q[n-1](t,s); Q[1](t,s) = ts. - Emeric Deutsch, Oct 29 2006

Corrected and extended by R. J. Mathar, Feb 05 2007

Entry revised by N. J. A. Sloane, Jun 01 2005, Jun 16 2007

A113547 Triangle read by rows: number of labeled partitions of n with maximin m.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 4, 5, 5, 1, 8, 13, 15, 15, 1, 16, 35, 47, 52, 52, 1, 32, 97, 153, 188, 203, 203, 1, 64, 275, 515, 706, 825, 877, 877, 1, 128, 793, 1785, 2744, 3479, 3937, 4140, 4140, 1, 256, 2315, 6347, 11002, 15177, 18313, 20270, 21147, 21147, 1, 512, 6817, 23073, 45368, 68303, 88033, 102678, 111835, 115975, 115975

Offset: 1

Franklin T. Adams-Watters, Jan 13 2006

The maximin of a partition is the maximum over all parts of the minimum label in each part. If the rows are reversed, the result is the number of partitions of n with minimax m.

The number of restricted growth functions of length n where the maximum appears first at position m. The RGF's are defined here as f(1)=1 and f(i) <=1+max_{1<=jR. J. Mathar, Mar 18 2016

			Maximin [123]=max(1)=1, maximin [12|3]=max(1,3)=3, maximin [13|2]=max(1,2)=2, maximin [1|23]=max(1,2)=2 and maximin [1|2|3]=max(1,2,3)=3, so for n=3 the multiset of maximins is {1,2,2,3,3}, making the 3rd line 1,2,2.
1;
1,  1;
1,  2,   2;
1,  4,   5,   5;
1,  8,  13,  15,  15;
1, 16,  35,  47,  52,  52;
1, 32,  97, 153, 188, 203, 203;
1, 64, 275, 515, 706, 825, 877, 877;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Todd Tichenor, A note on graph compositions and their connection to minimax of set partitions, arXiv:1709.00393 [math.CO], 2017.

Cf. A008277, A000110, A271466.

A362924 and A362925 are other versions of this triangle. - N. J. A. Sloane, Aug 10 2023

A113547 := proc(n,m)
    add(combinat[stirling2](m-1,k-1)*k^(n-m),k=1..m) ;
end proc:
seq(seq( A113547(n,m),m=1..n),n=1..10) ; # R. J. Mathar, Mar 13 2016

T[n_, n_] := BellB[n - 1]; T[n_, n_ - 1] := BellB[n - 1]; T[n_, n_ - 2] := BellB[n - 1] - BellB[n - 3]; T[n_, m_] := Sum[StirlingS2[m - 1, k - 1]*k^(n - m), {k, 1, m}]; Table[T[n, m], {n, 1, 5}, {m, 1, n}] (* G. C. Greubel, May 06 2017 *)

T(n, m) = Sum_{k=1..m} S2(m-1, k-1)*k^(n-m), where S2 is the Stirling numbers of the second kind (A008277). T(n, n)=T(n, n-1)=B(n-1), where B is the Bell numbers (A000110). T(n, n-2)=B(n-1)-B(n-3).

Conjectures: T(n,3) = A007689(n-3). T(n,4) = 2^(n-4)+3^(n-3)+4^(n-4).- R. J. Mathar, Mar 13 2016

A271467 Number of set partitions of [2n] such that n is the largest element of the last block.

Original entry on oeis.org

0, 1, 10, 149, 3149, 88772, 3195161, 142084077, 7611839904, 481614364251, 35417157810681, 2987457716884346, 285867917955683723, 30741794345071162069, 3685435185993122202106, 489101076818110701758097, 71414666988339306082779537, 11410190978037926887950636172

Offset: 1

Alois P. Heinz, Apr 08 2016

nonn

Each set partition is written as a sequence of blocks, ordered by the smallest elements in the blocks.

			a(2) = 1: 134|2.
a(3) = 10: 12456|3, 1456|23, 1456|2|3, 145|26|3, 146|25|3, 14|256|3, 156|24|3, 15|246|3, 16|245|3, 1|2456|3.

Alois P. Heinz, Table of n, a(n) for n = 1..200
Wikipedia, Partition of a set

Cf. A271466.

a(n) = A271466(2n,n).

A271607 Number of set partitions of [2n+1] such that n+1 is the largest element of the last block.

Original entry on oeis.org

1, 1, 6, 63, 1039, 24190, 745107, 29058813, 1389893708, 79588371929, 5353760622719, 416660175523064, 37047640989016445, 3724084616168887373, 419437505978046355690, 52523298180976612585435, 7263669823685446959438851, 1102849583101324096499809166

Offset: 0

Alois P. Heinz, Apr 10 2016

nonn

Each set partition is written as a sequence of blocks, ordered by the smallest elements in the blocks.

			a(0) = 1: 1.
a(1) = 1: 13|2.
a(2) = 6: 1245|3, 145|23, 145|2|3, 14|25|3, 15|24|3, 1|245|3.
a(3) = 63: 123567|4, 12567|34, 12567|3|4, 1256|37|4, ..., 1|26|357|4, 17|2|356|4, 1|27|356|4, 1|2|3567|4.

Alois P. Heinz, Table of n, a(n) for n = 0..200
Wikipedia, Partition of a set

Cf. A271466.

a(n) = A271466(2n+1,n+1).

A271743 Number of set partitions of [n] such that 4 is the largest element of the last block.

Original entry on oeis.org

10, 15, 29, 63, 149, 375, 989, 2703, 7589, 21735, 63149, 185343, 547829, 1627095, 4848509, 14479983, 43308869, 129664455, 388469069, 1164358623, 3490978709, 10468741815, 31397836829, 94176733263, 282496645349, 847422827175, 2542134263789, 7626134355903

Offset: 4

Alois P. Heinz, Apr 13 2016

			a(4) = 10: 1234, 123|4, 12|34, 12|3|4, 13|24, 13|2|4, 1|234, 1|23|4, 1|2|34, 1|2|3|4.

Alois P. Heinz, Table of n, a(n) for n = 4..1000
Wikipedia, Partition of a set
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Column k=4 of A271466.

I:=[10,15,29,63]; [n le 4 select I[n] else 6*Self(n-1) -11*Self(n-2)+6*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Apr 13 2016

Join[{10},LinearRecurrence[{6,-11,6},{15,29,63},30]] (* Vincenzo Librandi, Apr 13 2016 *)

Vec(x^4*(2 - 3*x)*(5 - 15*x + 2*x^2) / ((1 - x)*(1 - 2*x)*(1 - 3*x)) + O(x^40)) \\ Colin Barker, May 21 2017

G.f.: x^4*(3*x-2)*(2*x^2-15*x+5)/Product_{j=1..3} (j*x-1).
From Colin Barker, May 21 2017: (Start)
G.f.: x^4*(2 - 3*x)*(5 - 15*x + 2*x^2) / ((1 - x)*(1 - 2*x)*(1 - 3*x)).
a(n) = 4 + 2^(n-2) + 3^(n-4) for n>4.
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n>7.
(End)

A271744 Number of set partitions of [n] such that 5 is the largest element of the last block.

Original entry on oeis.org

30, 59, 139, 365, 1039, 3149, 10039, 33365, 114799, 406589, 1475239, 5460965, 20550559, 78375629, 302129239, 1174610165, 4597146319, 18085554269, 71434828039, 283020552965, 1123932750079, 4471296246509, 17811907987639, 71027855481365, 283452201569839

Offset: 5

Alois P. Heinz, Apr 13 2016

Alois P. Heinz, Table of n, a(n) for n = 5..1000
Wikipedia, Partition of a set
Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Column k=5 of A271466.

I:=[30,59,139,365,1039]; [n le 5 select I[n] else 10*Self(n-1)-35*Self(n-2)+50*Self(n-3)-24*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Apr 13 2016

Join[{30}, LinearRecurrence[{10, -35, 50, -24}, {59, 139, 365, 1039}, 30]] (* Vincenzo Librandi, Apr 13 2016 *)

Vec(x^5*(30 - 241*x + 599*x^2 - 460*x^3 + 24*x^4) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)) + O(x^40)) \\ Colin Barker, Dec 24 2017

G.f.: x^5*(24*x^4-460*x^3+599*x^2-241*x+30)/Product_{j=1..4} (j*x-1).
From Colin Barker, Dec 24 2017: (Start)
a(n) = 8 + 13*2^(n-5) + 7*3^(n-5) + 4^(n-5) for n>5.
a(n) = 10*a(n-1) - 35*a(n-2) + 50*a(n-3) - 24*a(n-4) for n>9.
(End)

A271745 Number of set partitions of [n] such that 6 is the largest element of the last block.

Original entry on oeis.org

104, 250, 692, 2110, 6932, 24190, 88772, 340030, 1351412, 5546110, 23407652, 101247550, 447454292, 2015029630, 9224364932, 42832260670, 201341787572, 956443162750, 4584181712612, 22137843427390, 107584138943252, 525581866073470, 2578798342362692

Offset: 6

Alois P. Heinz, Apr 13 2016

Alois P. Heinz, Table of n, a(n) for n = 6..1000
Wikipedia, Partition of a set
Index entries for linear recurrences with constant coefficients, signature (15,-85,225,-274,120).

Column k=6 of A271466.

I:=[104,250,692,2110,6932,24190]; [n le 6 select I[n] else 15*Self(n-1)-85*Self(n-2)+225*Self(n-3)-274*Self(n-4)+120*Self(n-5): n in [1..30]]; // Vincenzo Librandi, Apr 14 2016

Join[{104}, LinearRecurrence[{15, -85, 225, -274, 120}, {250, 692, 2110, 6932, 24190}, 30]] (* Vincenzo Librandi, Apr 14 2016 *)

x='x+O('x^99); Vec(2*x^6*(60*x^5-3174*x^4+5210*x^3-2891*x^2+655*x-52) / (120*x^5-274*x^4+225*x^3-85*x^2+15*x-1)) \\ Altug Alkan, Apr 14 2016

G.f.: 2*x^6*(60*x^5-3174*x^4+5210*x^3-2891*x^2+655*x-52) / Product_{j=1..5} (j*x-1).
From Colin Barker, Jan 04 2018: (Start)
a(n) = 16 + 5*2^(n-3) + 35*3^(n-6) + 11*4^(n-6) + 5^(n-6) for n>6.
a(n) = 15*a(n-1) - 85*a(n-2) + 225*a(n-3) - 274*a(n-4) + 120*a(n-5) for n>11.
(End)

A271746 Number of set partitions of [n] such that 7 is the largest element of the last block.

Original entry on oeis.org

406, 1145, 3627, 12521, 46299, 181265, 745107, 3195161, 14220459, 65412065, 309878787, 1507297001, 7508078619, 38208764465, 198238593267, 1046593626041, 5612793712779, 30528112814465, 168152752952547, 936705967782281, 5270538854994939, 29919810501018065

Offset: 7

Alois P. Heinz, Apr 13 2016

Alois P. Heinz, Table of n, a(n) for n = 7..1000
Wikipedia, Partition of a set
Index entries for linear recurrences with constant coefficients, signature (21,-175,735,-1624,1764,-720).

Column k=7 of A271466.

LinearRecurrence[{21,-175,735,-1624,1764,-720},{406,1145,3627,12521,46299,181265,745107},30] (* Harvey P. Dale, Jun 12 2022 *)

Vec(x^7*(406 - 7381*x + 50632*x^2 - 161681*x^3 + 235852*x^4 - 122388*x^5 + 720*x^6) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)) + O(x^40)) \\ Colin Barker, Jan 04 2018

G.f.: x^7*(720*x^6-122388*x^5+235852*x^4-161681*x^3+50632*x^2-7381*x+406) / Product_{j=1..6} (j*x-1).
From Colin Barker, Jan 04 2018: (Start)
a(n) = 32 + 121*2^(n-7) + 155*3^(n-7) + 5*4^(n-5) + 16*5^(n-7) + 6^(n-7) for n>7.
a(n) = 21*a(n-1) - 175*a(n-2) + 735*a(n-3) - 1624*a(n-4) + 1764*a(n-5) - 720*a(n-6) for n>13.
(End)

A271747 Number of set partitions of [n] such that 8 is the largest element of the last block.

Original entry on oeis.org

1754, 5649, 20085, 77133, 315597, 1362669, 6164685, 29058813, 142084077, 717966669, 3737612685, 19991467293, 109605434157, 614681711469, 3519553748685, 20540447808573, 121996580169837, 736352527581069, 4510823754140685, 28011087761890653, 176122939449075117

Offset: 8

Alois P. Heinz, Apr 13 2016

Alois P. Heinz, Table of n, a(n) for n = 8..1000
Wikipedia, Partition of a set
Index entries for linear recurrences with constant coefficients, signature (28,-322,1960,-6769,13132,-13068,5040).

Column k=8 of A271466.

Vec(x^8*(1754 - 43463*x + 426701*x^2 - 2104109*x^3 + 5424029*x^4 - 6799268*x^5 + 3145476*x^6 - 5040*x^7) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)) + O(x^40)) \\ Colin Barker, Jan 04 2018

G.f.: x^8 *(5040*x^7 -3145476*x^6 +6799268*x^5 -5424029*x^4 +2104109*x^3 -426701*x^2 +43463*x -1754)/Product_{j=1..7} (j*x-1).
From Colin Barker, Jan 04 2018: (Start)
a(n) = 64 + 91*2^(n-6) + 245*2^(2*n-15) + 11*2^(n-7)*3^(n-8) + 217*3^(n-7) + 161*5^(n-8) + 7^(n-8) for n>8.
a(n) = 28*a(n-1) - 322*a(n-2) + 1960*a(n-3) - 6769*a(n-4) + 13132*a(n-5) - 13068*a(n-6) + 5040*a(n-7) for n>15.
(End)

cp's OEIS Frontend

A186021 a(n) = Bell(n)*(2 - 0^n).

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A095149 Triangle read by rows: Aitken's array (A011971) but with a leading diagonal before it given by the Bell numbers (A000110), 1, 1, 2, 5, 15, 52, ...

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A113547 Triangle read by rows: number of labeled partitions of n with maximin m.

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A271467 Number of set partitions of [2n] such that n is the largest element of the last block.

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A271607 Number of set partitions of [2n+1] such that n+1 is the largest element of the last block.

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A271743 Number of set partitions of [n] such that 4 is the largest element of the last block.

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A271744 Number of set partitions of [n] such that 5 is the largest element of the last block.

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