A276924 -id:A276924 - cp's OEIS frontend

A001681 The partition function G(n,4).

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

A276921 Number A(n,k) of ordered set partitions of [n] with at most k elements per block; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 3, 6, 0, 1, 1, 3, 12, 24, 0, 1, 1, 3, 13, 66, 120, 0, 1, 1, 3, 13, 74, 450, 720, 0, 1, 1, 3, 13, 75, 530, 3690, 5040, 0, 1, 1, 3, 13, 75, 540, 4550, 35280, 40320, 0, 1, 1, 3, 13, 75, 541, 4670, 45570, 385560, 362880, 0

Offset: 0

Alois P. Heinz, Sep 22 2016

			Square array A(n,k) begins:
  1,    1,     1,     1,     1,     1,     1,     1, ...
  0,    1,     1,     1,     1,     1,     1,     1, ...
  0,    2,     3,     3,     3,     3,     3,     3, ...
  0,    6,    12,    13,    13,    13,    13,    13, ...
  0,   24,    66,    74,    75,    75,    75,    75, ...
  0,  120,   450,   530,   540,   541,   541,   541, ...
  0,  720,  3690,  4550,  4670,  4682,  4683,  4683, ...
  0, 5040, 35280, 45570, 47110, 47278, 47292, 47293, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Daniel Birmajer, Juan B. Gil, David S. Kenepp, and Michael D. Weiner, Restricted generating trees for weak orderings, arXiv:2108.04302 [math.CO], 2021.

Columns k=0..10 give: A000007, A000142, A080599, A189886, A276924, A276925, A276926, A276927, A276928, A276929, A276930.
Main diagonal gives A000670.
Cf. A276922.

A:= proc(n, k) option remember; `if`(n=0, 1, add(
       A(n-i, k)*binomial(n, i), i=1..min(n, k)))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

A[n_, k_] := A[n, k] = If[n==0, 1, Sum[A[n-i, k]*Binomial[n, i], {i, 1, Min[n, k]}]]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 03 2017, translated from Maple *)

E.g.f. of column k: 1/(1-Sum_{i=1..k} x^i/i!).

A(n,k) = Sum_{j=0..k} A276922(n,j).

A320760 Number of ordered set partitions of [n] where the maximal block size equals four.

Original entry on oeis.org

1, 10, 120, 1540, 21490, 326970, 5402250, 96500250, 1855334250, 38228190000, 840776937000, 19666511865000, 487617137007000, 12776791730703000, 352825452012033000, 10242418813814187000, 311854958169459705000, 9937942309809373860000, 330821844137019184950000

Offset: 4

Alois P. Heinz, Oct 20 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..424

Column k=4 of A276922.

Cf. A189886, A276924.

b:= proc(n, k) option remember; `if`(n=0, 1, add(
      b(n-i, k)*binomial(n, i), i=1..min(n, k)))
    end:
a:= n-> (k-> b(n, k) -b(n, k-1))(4):
seq(a(n), n=4..25);

b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[b[n - i, k] Binomial[n, i], {i, 1, Min[n, k]}]];
a[n_] := With[{k = 4}, b[n, k] - b[n, k-1]];
a /@ Range[4, 25] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)

E.g.f.: 1/(1-Sum_{i=1..4} x^i/i!) - 1/(1-Sum_{i=1..3} x^i/i!).

a(n) = A276924(n) - A189886(n).

A320761 Number of ordered set partitions of [n] where the maximal block size equals five.

Original entry on oeis.org

1, 12, 168, 2464, 38808, 657972, 11997216, 234594360, 4903616718, 109205019924, 2582909885556, 64686057980544, 1710536977653504, 47637803779229664, 1393903719674129664, 42758329987344875904, 1372254504736418142840, 45989719374155059863360

Offset: 5

Alois P. Heinz, Oct 20 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..424

Column k=5 of A276922.

Cf. A276924, A276925.

b:= proc(n, k) option remember; `if`(n=0, 1, add(
      b(n-i, k)*binomial(n, i), i=1..min(n, k)))
    end:
a:= n-> (k-> b(n, k) -b(n, k-1))(5):
seq(a(n), n=5..25);

b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[b[n - i, k] Binomial[n, i], {i, 1, Min[n, k]}]];
a[n_] := With[{k = 5}, b[n, k] - b[n, k-1]];
a /@ Range[5, 25] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)

E.g.f.: 1/(1-Sum_{i=1..5} x^i/i!) - 1/(1-Sum_{i=1..4} x^i/i!).

a(n) = A276925(n) - A276924(n).

A355294 Expansion of e.g.f. 1 / (1 - x - x^2/2 - x^3/3 - x^4/4).

Original entry on oeis.org

1, 1, 3, 14, 88, 670, 6170, 66360, 815640, 11272800, 173132400, 2925014400, 53909394000, 1076365290000, 23144112591600, 533193460800000, 13102608591072000, 342105146182800000, 9457689380931792000, 275988880808825184000, 8477631163592791200000, 273430368958004818560000, 9238944655686318693120000

Offset: 0

Ilya Gutkovskiy, Jun 27 2022

nonn

Cf. A007840, A070945, A080599, A276924, A355293.

nmax = 22; CoefficientList[Series[1/(1 - x - x^2/2 - x^3/3 - x^4/4), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = a[1] = 1; a[2] = 3; a[3] = 14; a[n_] := a[n] = n a[n - 1] + n (n - 1) a[n - 2]/2 + n (n - 1) (n - 2) a[n - 3]/3 + n (n - 1) (n - 2) (n - 3) a[n - 4]/4; Table[a[n], {n, 0, 22}]

a(n) = n * a(n-1) + n * (n-1) * a(n-2) / 2 + n * (n-1) * (n-2) * a(n-3) / 3 + n * (n-1) * (n-2) * (n-3) * a(n-4) / 4.

A365293 a(n) = n!*tetranacci(n+3).

Original entry on oeis.org

1, 1, 4, 24, 192, 1800, 20880, 282240, 4354560, 75479040, 1455148800, 30855686400, 713712384000, 17884003737600, 482619020083200, 13954193180928000, 430360865206272000, 14102295149150208000, 489295008086556672000, 17919783031425859584000

Offset: 0

Enrique Navarrete, Aug 31 2023

nonn

a(n) is the number of ways to partition [n] into blocks of size at most 4, order the blocks, and order the elements within each block.

			a(5) = 1800 since the number of ways to partition [5] into blocks of size at most 4, order the blocks, and order the elements within each block are the following:
1) 1234,5: 10 such ordered blocks; 240 ways;
2) 123,4,5: 60 such ordered blocks; 360 ways;
3) 123,45: 20 such ordered blocks; 240 ways;
4) 12,34,5: 90 such ordered blocks; 360 ways;
5) 12,3,4,5: 240 such ordered blocks; 480 ways;
6) 1,2,3,4,5: 120 such ordered blocks; 120 ways.

Cf. A000078, A000142, A002866, A005442, A276924, A364324.

Table[n! SeriesCoefficient[1/(1-x-x^2-x^3-x^4),{x,0,n}],{n,0,19}] (* Stefano Spezia, Aug 31 2023 *)

E.g.f.: 1/(1-x-x^2-x^3-x^4).

a(n) = A000142(n) * A000078(n+3).

cp's OEIS Frontend

A001681 The partition function G(n,4).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A276921 Number A(n,k) of ordered set partitions of [n] with at most k elements per block; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A320760 Number of ordered set partitions of [n] where the maximal block size equals four.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A320761 Number of ordered set partitions of [n] where the maximal block size equals five.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A355294 Expansion of e.g.f. 1 / (1 - x - x^2/2 - x^3/3 - x^4/4).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A365293 a(n) = n!*tetranacci(n+3).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula