A229223 -id:A229223 - cp's OEIS frontend

A080510 Triangle read by rows: T(n,k) gives the number of set partitions of {1,...,n} with maximum block length k.

1, 1, 1, 1, 3, 1, 1, 9, 4, 1, 1, 25, 20, 5, 1, 1, 75, 90, 30, 6, 1, 1, 231, 420, 175, 42, 7, 1, 1, 763, 2016, 1015, 280, 56, 8, 1, 1, 2619, 10024, 6111, 1890, 420, 72, 9, 1, 1, 9495, 51640, 38010, 12978, 3150, 600, 90, 10, 1, 1, 35695, 276980, 244035, 91938, 24024, 4950, 825, 110, 11, 1

Offset: 1

Wouter Meeussen, Mar 22 2003

Row sums are A000110 (Bell numbers). Second column is A001189 (Degree n permutations of order exactly 2).
From Peter Luschny, Mar 09 2009: (Start)
Partition product of Product_{j=0..n-1} ((k + 1)*j - 1) and n! at k = -1, summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A036040.
Same partition product with length statistic is A008277.
Diagonal a(A000217) = A000012.
Row sum is A000110. (End)
From Gary W. Adamson, Feb 24 2011: (Start)
Construct an array in which the n-th row is the partition function G(n,k), where G(n,1),...,G(n,6) = A000012, A000085, A001680, A001681, A110038, A148092, with the first few rows
  1,   1,   1,   1,   1,   1,    1, ... = A000012
  1,   2,   4,  10,  26,  76,  232, ... = A000085
  1,   2,   5,  14,  46, 166,  652, ... = A001680
  1,   2,   5,  15,  51, 196,  827, ... = A001681
  1,   2    5   15   52  202   869, ... = A110038
  1,   2,   5   15   52  203   876, ... = A148092
  ...
Rows tend to A000110, the Bell numbers. Taking finite differences from the top, then reorienting, we obtain triangle A080510.
The n-th row of the array is the eigensequence of an infinite lower triangular matrix with n diagonals of Pascal's triangle starting from the right and the rest zeros. (End)

			T(4,3) = 4 since there are 4 set partitions with longest block of length 3: {{1},{2,3,4}}, {{1,3,4},{2}}, {{1,2,3},{4}} and {{1,2,4},{3}}.
Triangle begins:
  1;
  1,    1;
  1,    3,     1;
  1,    9,     4,    1;
  1,   25,    20,    5,    1;
  1,   75,    90,   30,    6,   1;
  1,  231,   420,  175,   42,   7,  1;
  1,  763,  2016, 1015,  280,  56,  8,  1;
  1, 2619, 10024, 6111, 1890, 420, 72,  9,  1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Peter Luschny, Counting with Partitions.
Peter Luschny, Generalized Stirling_2 Triangles.
J. Riordan, Letter, 11/23/1970. See second page of letter.

Cf. A080107, A080337, A008277, A178979, A276922, A327884.
Cf. A157396, A157397, A157398, A157399, A157400, A157401, A157402, A157403, A157404, A157405. - Peter Luschny, Mar 09 2009
Cf. A000012, A000085, A001680, A001681, A110038, A148092. - Gary W. Adamson, Feb 24 2011
Columns k=1..10 give: A000012 (for n>0), A001189, A229245, A229246, A229247, A229248, A229249, A229250, A229251, A229252. - Alois P. Heinz, Sep 17 2013
T(2n,n) gives A276961.
Take differences along rows of A229223. - N. J. A. Sloane, Jan 10 2018

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       add(b(n-i*j, i-1) *n!/i!^j/(n-i*j)!/j!, j=0..n/i)))
    end:
T:= (n, k)-> b(n, k) -b(n, k-1):
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Apr 20 2012

<< DiscreteMath`NewCombinatorica`; Table[Length/@Split[Sort[Max[Length/@# ]&/@SetPartitions[n]]], {n, 12}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[b[n-i*j, i-1]*n!/i!^j/(n-i*j)!/j!, {j, 0, n/i}]]]; T[n_, k_] := b[n, k]-b[n, k-1]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Feb 25 2014, after Alois P. Heinz *)

E.g.f. for k-th column: exp(exp(x)*GAMMA(k, x)/(k-1)!-1)*(exp(x^k/k!)-1). - Vladeta Jovovic, Feb 04 2005
From Peter Luschny, Mar 09 2009: (Start)
T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n.
T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,...,a_n such that
1*a_1 + 2*a_2 + ... + n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*...*a_n!),
f^a = (f_1/1!)^a_1*...*(f_n/n!)^a_n and f_n = Product_{j=0..n-1} (-1) = (-1)^n. (End)
From Ludovic Schwob, Jan 15 2022: (Start)
T(2n,n) = C(2n,n)*(A000110(n)-1/2) for n>0.
T(n,m) = C(n,m)*A000110(n-m) for 2m > n > 0. (End)

A001680 The partition function G(n,3).

Original entry on oeis.org

1, 1, 2, 5, 14, 46, 166, 652, 2780, 12644, 61136, 312676, 1680592, 9467680, 55704104, 341185496, 2170853456, 14314313872, 97620050080, 687418278544, 4989946902176, 37286121988256, 286432845428192, 2259405263572480, 18280749571449664, 151561941235370176

Offset: 0

N. J. A. Sloane

nonn

Number of '12-3 and 21-3'-avoiding permutations.
Set partitions into sets of size at most 3. The e.g.f. for partitions into sets of size at most s is exp( sum(j=1..s, x^j/j!) ). [Joerg Arndt, Dec 07 2012]
Also called restricted Stirling numbers of the second kind (see Mezo). - N. J. A. Sloane, Nov 27 2013

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).

Seiichi Manyama, Table of n, a(n) for n = 0..653 (terms 0..250 from Alois P. Heinz)
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394 [math.CO], 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Filippo Disanto and Thomas Wiehe, Some instances of a sub-permutation problem on pattern avoiding permutations, arXiv preprint arXiv:1210.6908 [math.CO], 2012.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 18
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
T. Mansour, Restricted permutations by patterns of type 2-1, arXiv:math/0202219 [math.CO], 2002.
I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013 and J. Int. Seq. 17 (2014) #14.1.1.
F. L. Miksa, L. Moser and M. Wyman, Restricted partitions of finite sets, Canad. Math. Bull., 1 (1958), 87-96.

Cf. A001681, A189886.

Column k=3 of A229223.

G:= proc(n, i) option remember;
      `if`(n=0, 1, `if`(i<1, 0,
       add(G(n-i*j, i-1) *n!/i!^j/(n-i*j)!/j!, j=0..n/i)))
    end:
a:= n-> G(n, 3):
seq(a(n), n=0..30); # Alois P. Heinz, Apr 20 2012
# Recurrence:
rec := {(-n^2-3*n-2)*f(n)+(-2*n-4)*f(n+1)-2*f(n+2)+2*f(n+3)=0,f(0)=1,f(1)=1,f(2)=2}:
aList := gfun:-rectoproc(rec,f(n),list): aList(25); # Peter Luschny, Feb 26 2018

Table[Sum[n!/(m!2^(n+j-2m)3^(m-j))Binomial[m,j]Binomial[j,n+2j-3m],{m,0,n},{j,0,3m-n}],{n,0,15}]

E.g.f.: exp ( x + x^2 / 2 + x^3 / 6 ).
a(n) = n! * sum(k=1..n, 1/k! * sum(j=0..k, C(k,j) * C(j,n-3*k+2*j) * 2^(-n+2*k-j) * 3^(j-k))). [Vladimir Kruchinin, Jan 25 2011]
a(n) = G(n,3) with G(0,i) = 1, G(n,i) = 0 for n>0 and i<1, otherwise G(n,i) = Sum_{j=0..floor(n/i)} G(n-i*j,i-1) * n!/(i!^j*(n-i*j)!*j!). - Alois P. Heinz, Apr 20 2012
D-finite with recurrence 2*a(n) -2*a(n-1) +2*(-n+1)*a(n-2) -(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Jan 25 2013
Proof of foregoing recurrence: The partition containing n can be a singleton (a(n-1) partitions of the remaining terms), a doubleton ((n-1) choices for its companion times a(n-2) partitions of the remaining terms) or a tripleton ((n-1) choose 2 choices for its companions times a(n-3) partitions for the remaining terms), so a(n) = a(n-1) + (n-1)a(n-2) + (n-1)*(n-2)/2 * a(n-3). - Micah E. Fogel, Feb 14 2013
a(n) ~ n^(2*n/3)*exp(1/2*(2*n)^(2/3)+2/3*(2*n)^(1/3)-2*n/3-4/9)/(sqrt(3)*2^(n/3)). - Vaclav Kotesovec, May 29 2013

More terms added May 13 2009

A001681 The partition function G(n,4).

Original entry on oeis.org

1, 1, 2, 5, 15, 51, 196, 827, 3795, 18755, 99146, 556711, 3305017, 20655285, 135399720, 927973061, 6631556521, 49294051497, 380306658250, 3039453750685, 25120541332271, 214363100120051, 1885987611214092, 17085579637664715, 159185637725413675

Offset: 0

N. J. A. Sloane

Number of '12-3 and 321-4'-avoiding permutations.
Set partitions into sets of size at most 4. The e.g.f. for partitions into sets of size at most s is exp( sum(j=1..s, x^j/j!) ). [Joerg Arndt, Dec 07 2012]
Also called restricted Stirling numbers of the second kind (see Mezo). - N. J. A. Sloane, Nov 27 2013

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).

Seiichi Manyama, Table of n, a(n) for n = 0..609 (terms 0..200 from Alois P. Heinz)
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394 [math.CO], 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Filippo Disanto and Thomas Wiehe, Some instances of a sub-permutation problem on pattern avoiding permutations, arXiv preprint arXiv:1210.6908 [math.CO], 2012.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 19
Vladimir Victorovich Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
T. Mansour, Restricted permutations by patterns of type 2-1, arXiv:math/0202219 [math.CO], 2002.
I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013 and J. Int. Seq. 17 (2014) #14.1.1 .
F. L. Miksa, L. Moser and M. Wyman, Restricted partitions of finite sets, Canad. Math. Bull., 1 (1958), 87-96.

Cf. A001680, A276924.

Column k=4 of A229223.

G:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       add(G(n-i*j, i-1) *n!/i!^j/(n-i*j)!/j!, j=0..n/i)))
    end:
a:= n-> G(n, 4):
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 20 2012
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
       a(n-i)*binomial(n-1, i-1), i=1..min(n, 4)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 22 2016
# Recurrence:
rec := {(-n^3-6*n^2-11*n-6)*f(n) + (-3*n^2-15*n-18)*f(n+1) + (-6*n-18)*f(n+2) - 6*f(n+3) + 6*f(n+4)=0, f(0)=1, f(1)=1, f(2)=2, f(3)=5}:
aList := gfun:-rectoproc(rec, f(n), list): aList(24); # Peter Luschny, Feb 26 2018

g[n_, k_] := g[n, k] = If[n == 0, 1, If[k<1, 0, Sum[g[n-k*j, k-1]*n!/k!^j/(n-k*j)!/j!, {j, 0, n/k}]]]; Table[g[n, 4], {n, 0, 24}] (* Jean-François Alcover, Mar 11 2014, after Alois P. Heinz *)

A001681(n)=n!*sum(k=1,n, 1/k!*sum(j=0,k, binomial(k,j)*sum(i=j,n-k+j, binomial(j,i-j)*binomial(k-j,n-3*k+3*j-i)*2^(5*k-4*j+i-2*n)*3^(j-k))));
vector(33,n,A001681(n-1)) /* Joerg Arndt, Jan 25 2011 */

x='x+O('x^66); Vec(serlaplace(exp(sum(j=1,4,x^j/j!)))) \\ Joerg Arndt, Mar 11 2014

E.g.f.: exp( x + x^2/2 + x^3/6 + x^4/24 ). - Ralf Stephan, Apr 22 2004
a(n) = n! * sum(k=1..n, 1/k! * sum(j=0..k, C(k,j) * sum(i=j..n-k+j, C(j,i-j) * C(k-j,n-3*k+3*j-i) * 2^(5*k-4*j+i-2*n) * 3^(j-k)))). [Vladimir Kruchinin, Jan 25 2011]
a(n) = G(n,4) with G(0,i) = 1, G(n,i) = 0 for n>0 and i<1, otherwise G(n,i) = Sum_{j=0..floor(n/i)} G(n-i*j,i-1) * n!/(i!^j*(n-i*j)!*j!). - Alois P. Heinz, Apr 20 2012
Recurrence: 6*a(n) = 6*a(n-1) + 6*(n-1)*a(n-2) + 3*(n-2)*(n-1)*a(n-3) + (n-3)*(n-2)*(n-1)*a(n-4). - Vaclav Kotesovec, Sep 15 2013
a(n) ~ n^(3*n/4)*exp(31*(6*n)^(1/4)/64 + 5*sqrt(6*n)/16 + (6*n)^(3/4)/6 - 3*n/4 - 21/32)/(2*6^(n/4)) * (1 + 1599*6^(3/4)/(40960*n^(1/4)) + 280873603/1677721600*sqrt(6/n) + 33870741297579 /240518168576000 *6^(1/4)/n^(3/4)). - Vaclav Kotesovec, Sep 15 2013

More terms from Ralf Stephan, Apr 22 2004

A110038 The partition function G(n,5).

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 202, 869, 4075, 20645, 112124, 648649, 3976633, 25719630, 174839120, 1245131903, 9263053753, 71806323461, 578719497070, 4839515883625, 41916097982471, 375401824277096, 3471395994487422, 33099042344383885, 325005134436155395

Offset: 0

N. J. A. Sloane, May 13 2009

nonn

Set partitions into sets of size at most 5. The e.g.f. for partitions into sets of size at most s is exp( sum(j=1..s, x^j/j!) ). [Joerg Arndt, Dec 07 2012]

Seiichi Manyama, Table of n, a(n) for n = 0..591 (terms 0..200 from Alois P. Heinz)
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394 [math.CO], 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
F. L. Miksa, L. Moser and M. Wyman, Restricted partitions of finite sets, Canad. Math. Bull., 1 (1958), 87-96.

The sequences G(n,1), G(n,2), G(n,3), G(n,4), G(n,5), G(n,6) are given by A000012, A000085, A001680, A001681, A110038, A148092 respectively.
Column k=5 of A229223.
Cf. A276925.

G:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       add(G(n-i*j, i-1) *n!/i!^j/(n-i*j)!/j!, j=0..n/i)))
    end:
a:= n-> G(n, 5):
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 20 2012
# second Maple program:
a:= proc(n) option remember; `if`(n<5, [1, 1, 2, 5, 15][n+1],
      a(n-1)+(n-1)*(a(n-2)+(n-2)/2*(a(n-3)+(n-3)/3*(a(n-4)
      +(n-4)/4*a(n-5)))))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 15 2013

G[n_, i_] := G[n, i] = If[n == 0, 1, If[i<1, 0, Sum[G[n-i*j, i-1] *n!/i!^j/(n-i*j)!/j!, {j, 0, n/i}]]]; a[n_] := G[n, 5]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)

E.g.f.: exp( x + x^2/2 + x^3/6 + x^4/24 + x^5/120 ).
a(n) = n! * sum(k=1..n, 1/k! * sum(r=0..k, C(k,r) * sum(m=0..r, 2^(m-r) * C(r,m) * sum(j=0..m, C(m,j) * C(j,n-m-k-j-r) * 6^(j-m) * 24^(n-r-m-k-2*j) * 120^(m+k+j+r-n))))). - Vladimir Kruchinin, Jan 25 2011
a(n) = G(n,5) with G(0,i) = 1, G(n,i) = 0 for n>0 and i<1, otherwise G(n,i) = Sum_{j=0..floor(n/i)} G(n-i*j,i-1) * n!/(i!^j*(n-i*j)!*j!). - Alois P. Heinz, Apr 20 2012

A148092 The partition function G(n,6).

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 203, 876, 4131, 21065, 115274, 672673, 4163743, 27216840, 187160429, 1349511178, 10173555345, 79982663997, 654277037674, 5557624876513, 48931106059451, 445790174654588, 4196351007814659, 40757862664061104, 407944375184911787

Offset: 0

N. J. A. Sloane, May 13 2009

nonn

Set partitions into sets of size at most 6. The e.g.f. for partitions into sets of size at most s is exp( sum(j=1..s, x^j/j!) ). [Joerg Arndt, Dec 07 2012]

Alois P. Heinz, Table of n, a(n) for n = 0..500
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394 [math.CO], 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
F. L. Miksa, L. Moser and M. Wyman, Restricted partitions of finite sets, Canad. Math. Bull., 1 (1958), 87-96.

The sequences G(n,1), G(n,2), G(n,3), G(n,4), G(n,5), G(n,6) are given by A000012, A000085, A001680, A001681, A110038, A148092 respectively.
Column k=6 of A229223.
Cf. A276926.

G:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       add(G(n-i*j, i-1) *n!/i!^j/(n-i*j)!/j!, j=0..n/i)))
    end:
a:= n-> G(n, 6):
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 20 2012
# second Maple program:
a:= proc(n) option remember; `if`(n<6, [1, 1, 2, 5, 15, 52][n+1],
      a(n-1)+(n-1)*(a(n-2) +(n-2)/2*(a(n-3) +(n-3)/3*(a(n-4)
            +(n-4)/4*(a(n-5) +(n-5)/5*a(n-6))))))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 15 2013

G[n_, i_] := G[n, i] = If[n == 0, 1, If[i<1, 0, Sum[G[n-i*j, i-1] *n!/i!^j/(n-i*j)!/j!, {j, 0, n/i}]]]; a[n_] := G[n, 6]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)

E.g.f.: exp( x + x^2/2 + x^3/6 + x^4/24 + x^5/120 + x^6/720 ).

a(n) = G(n,6) with G(0,i) = 1, G(n,i) = 0 for n>0 and i<1, otherwise G(n,i) = Sum_{j=0..floor(n/i)} G(n-i*j,i-1) * n!/(i!^j*(n-i*j)!*j!). - Alois P. Heinz, Apr 20 2012

A229243 Number A(n,k) of set partitions of {1,...,k*n} into sets of size at most n; 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, 10, 5, 1, 1, 1, 76, 166, 15, 1, 1, 1, 764, 12644, 3795, 52, 1, 1, 1, 9496, 1680592, 3305017, 112124, 203, 1, 1, 1, 140152, 341185496, 6631556521, 1245131903, 4163743, 877, 1

Offset: 0

Alois P. Heinz, Sep 17 2013

			Square array A(n,k) begins:
  1,  1,      1,          1,              1,                   1, ...
  1,  1,      1,          1,              1,                   1, ...
  1,  2,     10,         76,            764,                9496, ...
  1,  5,    166,      12644,        1680592,           341185496, ...
  1, 15,   3795,    3305017,     6631556521,      25120541332271, ...
  1, 52, 112124, 1245131903, 41916097982471, 3282701194678476257, ...

Alois P. Heinz, Antidiagonals n = 0..30, flattened

Columns k=0-3 give: A000012, A000110, A229228, A229413.
Rows n=0+1, 2-3 give: A000012, A066223, A229414.
Main diagonal gives: A229229.
Cf. A229223.

G:= proc(n, k) option remember; local j; if k>n then G(n, n)
      elif n=0 then 1 elif k<1 then 0 else G(n-k, k);
      for j from k-1 to 1 by -1 do %*(n-j)/j +G(n-j, k) od; % fi
    end:
A:= (n, k)-> G(n*k, n):
seq(seq(A(n, d-n), n=0..d), d=0..10);

G[n_, k_] := G[n, k] = Module[{j, g}, Which[k > n, G[n, n], n == 0, 1, k < 1, 0, True, g = G[n-k, k]; For[j = k-1, j >= 1, j--, g = g*(n-j)/j + G[n-j, k] ]; g ] ]; A[n_, k_] := G[n*k, n]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 23 2013, translated from Maple *)

A(n,k) = (n*k)! * [x^(n*k)] exp(Sum_{j=1..n} x^j/j!).

A(n,k) = A229223(n*k,n).

A293051 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(Sum_{i=0..k} x^i/i! - exp(x)).

Original entry on oeis.org

1, 1, -1, 1, 0, 0, 1, 0, -1, 1, 1, 0, 0, -1, 1, 1, 0, 0, -1, 2, -2, 1, 0, 0, 0, -1, 9, -9, 1, 0, 0, 0, -1, -1, 9, -9, 1, 0, 0, 0, 0, -1, 9, -50, 50, 1, 0, 0, 0, 0, -1, -1, 34, -267, 267, 1, 0, 0, 0, 0, 0, -1, -1, 90, -413, 413, 1, 0, 0, 0, 0, 0, -1, -1, 34, -71

Offset: 0

Seiichi Manyama, Sep 29 2017

			Square array begins:
    1,  1,  1,  1,  1, ...
   -1,  0,  0,  0,  0, ...
    0, -1,  0,  0,  0, ...
    1, -1, -1,  0,  0, ...
    1,  2, -1, -1,  0, ...
   -2,  9, -1, -1, -1, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..4 give A000587, A293037, A293038, A293039, A293040.
Rows n=0..1 give A000012, (-1)*A000007.
Main diagonal gives A000007.
Cf. A229223, A293024.

E.g.f. of column k: Product_{i>k} exp(-x^i/i!).

A(0,k) = 1, A(1,k) = A(2,k) = ... = A(k,k) = 0 and A(n,k) = - Sum_{i=k..n-1} binomial(n-1,i)*A(n-1-i,k) for n > k.

A066223 Bisection of A000085.

Original entry on oeis.org

1, 2, 10, 76, 764, 9496, 140152, 2390480, 46206736, 997313824, 23758664096, 618884638912, 17492190577600, 532985208200576, 17411277367391104, 606917269909048576, 22481059424730751232, 881687990282453393920, 36494410645223834692096, 1589659519990672490875904

Offset: 0

N. J. A. Sloane, Dec 19 2001

Number of tableaux on 2n elements. - Roberto E. Martinez II, Jan 09 2002
a(n) = number of ways to connect 2n points labeled 1,2,...,2n in a line with 0 or more arcs such that at most one arc leaves each point. For example, with arcs separated by dashes, a(2)=10 counts {} (no arcs), 12, 13, 14, 23, 24, 34, 12-34, 13-24, 14-23. - David Callan, Sep 18 2007
a(n) = A229223(2n,2) = A229243(2,n). - Alois P. Heinz, Sep 17 2013

S. Chowla, The asymptotic behavior of solutions of difference equations, in Proceedings of the International Congress of Mathematicians (Cambridge, MA, 1950), Vol. I, 377, Amer. Math. Soc., Providence, RI, 1952.

Alois P. Heinz, Table of n, a(n) for n = 0..200
T. Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras
I. Dolinka, J. East and R. D. Gray, Motzkin monoids and partial Brauer monoids, arXiv preprint arXiv:1512.02279 [math.GR], 2015.
Michael Torpey, Semigroup congruences: computational techniques and theoretical applications, Ph.D. Thesis, University of St. Andrews (Scotland, 2019).

Cf. A066224.

a:= proc(n) option remember; `if`(n<2, n+1,
      (4*n-2)*a(n-1)-2*(n-1)*(2*n-3)*a(n-2))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 17 2013

NumberOfTableaux[2n]
a[n_] := a[n] = If[n<2, n+1, (4*n-2)*a[n-1] - 2*(n-1)*(2*n-3)*a[n-2]]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Oct 13 2014, after Alois P. Heinz *)
Table[(-2)^n HypergeometricU[-n, 1/2, -(1/2)], {n, 0, 90}] (* Emanuele Munarini, Aug 31 2017 *)

a(n)=sum(k=0,n,binomial(2*n,2*k)*prod(i=1,k,2*i-1))

a(n)=if(n<0, 0, n*=2; n!*polcoeff(exp(x+x^2/2+x*O(x^n)),n))

a(n) = sum(k=0, n, C(2n, 2*k)*(2k-1)!!). - Benoit Cloitre, May 01 2003
a(n) = n!*2^n*LaguerreL(n, -1/2, -1/2). - Vladeta Jovovic, May 10 2003
E.g.f.: cosh(x)*exp(x^2/2) (with interpolated zeros) - Paul Barry, May 26 2003
E.g.f.: exp(x/(1-2*x))/sqrt(1-2*x). - Paul Barry, Apr 12 2010
a(n) = (1/sqrt(2*pi))*Int((1+x)^(2*n)*exp(-x^2/2),x,-infinity,infinity). - Paul Barry, Apr 21 2010
Conjecture: a(n) +2*(-2*n+1)*a(n-1) +2*(n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Nov 24 2012
Remark: the above conjectured recurrence is true and can be obtained by the e.g.f. - Emanuele Munarini, Aug 31 2017
a(n) ~ n^n*2^(n-1/2)*exp(-n+sqrt(2*n)-1/4) * (1 + 7/(24*sqrt(2*n))). - Vaclav Kotesovec, Jun 22 2013

More terms from Roberto E. Martinez II, Jan 09 2002

A229224 The partition function G(n,7).

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 203, 877, 4139, 21137, 115874, 677623, 4204927, 27565188, 190168577, 1376119903, 10414950785, 82230347149, 675762947626, 5768465148493, 51054457464731, 467728049807348, 4428770289719931, 43281554035140829, 436015324638219779

Offset: 0

Alois P. Heinz, Sep 16 2013

nonn

Number G(n,7) of set partitions of {1,...,n} into sets of size at most 7.

Alois P. Heinz, Table of n, a(n) for n = 0..500

Column k=7 of A229223.

Cf. A276927.

G:= proc(n, k) option remember; local j; if k>n then G(n, n)
      elif n=0 then 1 elif k<1 then 0 else G(n-k, k);
      for j from k-1 to 1 by -1 do %*(n-j)/j +G(n-j, k) od; % fi
    end:
a:= n-> G(n, 7):
seq(a(n), n=0..30);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
       a(n-i)*binomial(n-1, i-1), i=1..min(n, 7)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 22 2016

CoefficientList[Exp[Sum[x^j/j!, {j, 1, 7}]] + O[x]^25, x]*Range[0, 24]! (* Jean-François Alcover, May 21 2018 *)

E.g.f.: exp(Sum_{j=1..7} x^j/j!).

A229225 The partition function G(n,8).

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 203, 877, 4140, 21146, 115964, 678448, 4212352, 27632112, 190778186, 1381763398, 10468226150, 82744297014, 680835331228, 5819712427654, 51584619782546, 473344099095848, 4489677962922186, 43957668431564086, 443694809361207824

Offset: 0

Alois P. Heinz, Sep 16 2013

nonn

Number G(n,8) of set partitions of {1,...,n} into sets of size at most 8.

Alois P. Heinz, Table of n, a(n) for n = 0..500

Column k=8 of A229223.

Cf. A276928.

G:= proc(n, k) option remember; local j; if k>n then G(n, n)
      elif n=0 then 1 elif k<1 then 0 else G(n-k, k);
      for j from k-1 to 1 by -1 do %*(n-j)/j +G(n-j, k) od; % fi
    end:
a:= n-> G(n, 8):
seq(a(n), n=0..30);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
       a(n-i)*binomial(n-1, i-1), i=1..min(n, 8)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 22 2016

CoefficientList[Exp[Sum[x^j/j!, {j, 1, 8}]] + O[x]^25, x]*Range[0, 24]! (* Jean-François Alcover, May 21 2018 *)

E.g.f.: exp(Sum_{j=1..8} x^j/j!).

cp's OEIS Frontend

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A293051 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(Sum_{i=0..k} x^i/i! - exp(x)).

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