A124421 -id:A124421 - cp's OEIS frontend

A108459 Number of labeled partitions of (n,n) into pairs (i,j).

1, 1, 5, 52, 855, 19921, 614866, 24040451, 1152972925, 66200911138, 4465023867757, 348383154017581, 31052765897026352, 3128792250765898965, 353179564583216567917, 44320731930172534543092, 6141797839043095806714667, 934330605640859569909566925

Offset: 0

Christian G. Bower, Jun 03 2005

nonn

Partitions of n black objects labeled 1..n and n white objects labeled 1..n. Each partition must have at least one white object.

a(n) is also the number of elements of the partition monoid P_n with domain {1,...,n}. Elements of P_n are set partitions of {1,1',...,n,n'}, and the domain of such a partition is the set of all points in {1,...,n} that belong to a block containing a dashed element. - James East, Apr 10 2018

Seiichi Manyama, Table of n, a(n) for n = 0..200

Main diagonal of A108458. Cf. A108461.
Cf. A048993 (Stirling2), A068424 (falling factorial).
Cf. A326600, A326270, A326271, A326288.
Bisection of A124421 (even part).

b:= proc(n) option remember; expand(`if`(n=0, 1,
      x*add(b(n-j)*binomial(n-1, j-1), j=1..n)))
    end:
a:= n-> add(coeff(b(n), x, j)*j^n, j=0..n):
seq(a(n), n=0..21);  # Alois P. Heinz, Dec 02 2023

a[n_] := If[n == 0, 1, Sum[k^n*StirlingS2[n, k], {k, 0, n}]];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Dec 10 2024 *)

{a(n)=polcoeff(sum(m=0, n, m^m*x^m/prod(k=1, m, 1-m*k*x +x*O(x^n))), n)} \\ Paul D. Hanna, Sep 17 2013

{a(n)=n!*polcoeff(sum(m=0, n, (exp(m*x+x*O(x^n))-1)^m/m!), n)} \\ Paul D. Hanna, Sep 17 2013

a(n) = Sum_{k=0..n} k^n*Stirling2(n,k). - Vladeta Jovovic, Aug 31 2006
E.g.f.: Sum_{n>=0} (exp(n*x)-1)^n / n!. - Vladeta Jovovic, Jul 12 2007
E.g.f.: Sum_{n>=0} exp(n^2*x) * exp( -exp(n*x) ) / n!. - Paul D. Hanna, Jun 28 2019
O.g.f.: Sum_{n>=0} n^n * x^n / Product_{k=1..n} (1 - n*k*x). - Paul D. Hanna, Sep 17 2013
a(n) = Sum_{k=0..n} Stirling2(n,k) * Sum_{l=k..n} Stirling2(n,l)*T(l,k). Here T(l,k) are the falling factorials. - James East, Apr 10 2018

A124419 Number of partitions of the set {1,2,...n} having no blocks that contain both odd and even entries.

Original entry on oeis.org

1, 1, 1, 2, 4, 10, 25, 75, 225, 780, 2704, 10556, 41209, 178031, 769129, 3630780, 17139600, 87548580, 447195609, 2452523325, 13450200625, 78697155750, 460457244900, 2859220516290, 17754399678409, 116482516809889, 764214897046969, 5277304280371714

Offset: 0

Emeric Deutsch, Oct 31 2006

nonn

			a(4) = 4 because we have 13|24, 1|24|3, 13|2|4 and 1|2|3|4.

Alois P. Heinz, Table of n, a(n) for n = 0..500
A. Dzhumadil’daev and D. Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764v1, 2014. [Version 1 contained many references to the OEIS, which were removed in Version 2. - _N. J. A. Sloane_, Mar 28 2015]
Askar Dzhumadil’daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.

Cf. A000110, A088911, A124418, A124420, A124421, A124422, A124423, A152875, A274310.
Column k=0 of A124418 and of A363493.
Column k=2 of A275069.

Q[0]:=1: for n from 1 to 30 do if n mod 2 = 1 then Q[n]:=expand(t*diff(Q[n-1],t)+x*diff(Q[n-1],s)+x*diff(Q[n-1],x)+t*Q[n-1]) else Q[n]:=expand(x*diff(Q[n-1],t)+s*diff(Q[n-1],s)+x*diff(Q[n-1],x)+s*Q[n-1]) fi od: for n from 0 to 30 do Q[n]:=Q[n] od: seq(subs({t=1,s=1,x=0},Q[n]),n=0..30);
# second Maple program:
with(combinat):
a:= n-> bell(floor(n/2))*bell(ceil(n/2)):
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 23 2013

a[n_] := BellB[Floor[n/2]]*BellB[Ceiling[n/2]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 20 2015, after Alois P. Heinz *)

a(n) = Q[n](1,1,0), where the polynomials Q[n]=Q[n](t,s,x) are defined by Q[0]=1; Q[n]=t*dQ[n-1]/dt + x*dQ[n-1]/ds + x*dQ[n-1]/dx + t*Q[n-1] if n is odd and Q[n]=x*dQ[n-1]/dt + s*dQ[n-1]/ds + x*dQ[n-1]/dx + s*Q[n-1] if n is even.
a(n) = A000110(floor(n/2)) * A000110(ceiling(n/2)). - Alois P. Heinz, Oct 23 2013
a(n) mod 2 = A088911(n). - Alois P. Heinz, Jun 06 2023

A124418 Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} having exactly k blocks that contain both odd and even entries (0<=k<=floor(n/2)).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 9, 2, 10, 30, 12, 25, 100, 72, 6, 75, 370, 372, 60, 225, 1369, 1922, 600, 24, 780, 5587, 9920, 4500, 360, 2704, 22801, 51200, 33750, 5400, 120, 10556, 101774, 273920, 234000, 55800, 2520, 41209, 454276, 1465472, 1622400, 576600, 52920, 720

Offset: 0

Emeric Deutsch, Oct 31 2006

Row n has 1+floor(n/2) terms. Row sums are the Bell numbers (A000110). T(n,0)=A124419(n).

			T(4,1) = 9 because we have 1234, 134|2, 1|234, 124|3, 14|2|3, 1|2|34, 123|4, 1|23|4 and 12|3|4.
Triangle starts:
   1;
   1;
   1,   1;
   2,   3;
   4,   9,  2;
  10,  30, 12;
  25, 100, 72, 6;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A000110, A124419, A124420, A124421, A124422, A124423.
Cf. A124526, A362495.
T(2n,n) gives A000142.

Q[0]:=1: for n from 1 to 13 do if n mod 2 = 1 then Q[n]:=expand(t*diff(Q[n-1],t)+x*diff(Q[n-1],s)+x*diff(Q[n-1],x)+t*Q[n-1]) else Q[n]:=expand(x*diff(Q[n-1],t)+s*diff(Q[n-1],s)+x*diff(Q[n-1],x)+s*Q[n-1]) fi od: for n from 0 to 13 do P[n]:=sort(subs({t=1,s=1},Q[n])) od: for n from 0 to 13 do seq(coeff(P[n],x,j),j=0..floor(n/2)) od; # yields sequence in triangular form
# second Maple program:
with(combinat):
T:= proc(n, k) local g, u; g:= floor(n/2); u:=ceil(n/2);
      add(binomial(g, i)*stirling2(i, k)*bell(g-i), i=k..g)*
      add(binomial(u, i)*stirling2(i, k)*bell(u-i), i=k..u)*k!
    end:
seq(seq(T(n,k), k=0..floor(n/2)), n=0..15); # Alois P. Heinz, Oct 23 2013

T[n_, k_] := Module[{g = Floor[n/2], u = Ceiling[n/2]}, Sum[Binomial[g, i] * StirlingS2[i, k]*BellB[g-i], {i, k, g}]*Sum[Binomial[u, i]*StirlingS2[i, k] * BellB[u-i], {i, k, u}]*k!]; Table[Table[T[n, k], {k, 0, Floor[n/2]}], {n, 0, 15}] // Flatten (* Jean-François Alcover, Feb 20 2015, after Alois P. Heinz *)

{T(n,k)=if(k<0||k>n,0, k!*(n\2)!*((n+1)\2)!*polcoeff(polcoeff(exp((1+y)*(exp(x+x*O(x^n))-1)),n\2),k) *polcoeff(polcoeff(exp((1+y)*(exp(x+x*O(x^n))-1)),(n+1)\2),k))} \\ Paul D. Hanna, Nov 08 2006

The generating polynomial of row n is P[n](x)=Q[n](1,1,x), where the polynomials Q[n]=Q[n](t,s,x) are defined by Q[0]=1; Q[n]=t*dQ[n-1]/dt + x*dQ[n-1]/ds + x*dQ[n-1]/dx + t*Q[n-1] if n is odd and Q[n]=x*dQ[n-1]/dt + s*dQ[n-1]/ds + x*dQ[n-1]/dx + s*Q[n-1] if n is even.
Conjecture: T(n,k) = k!*A049020([n/2],k)*A049020([(n+1)/2],k) where A049020(n,k)=Sum_{i=0..n} S2(n,i)*C(i,k) and S2(n,k)=(1/k!)*Sum_{j=0..k} (-1)^(k-j)*C(k,j)*j^n (the Stirling numbers of 2nd kind). - Paul D. Hanna, Nov 08 2006
Sum_{k=0..floor(n/2)} = k * A362495(n). - Alois P. Heinz, Jun 05 2023

A124420 Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n}, having exactly k blocks consisting only odd entries (0<=k<=ceiling(n/2)).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 3, 1, 5, 8, 2, 9, 26, 15, 2, 52, 101, 45, 5, 130, 385, 287, 70, 5, 855, 1889, 1143, 238, 15, 2707, 8295, 7320, 2475, 335, 15, 19921, 48382, 35805, 10540, 1275, 52, 75771, 240534, 240082, 100940, 19505, 1686, 52, 614866, 1609551, 1379753, 512710

Offset: 0

Emeric Deutsch, Oct 31 2006

Row n has 1 + ceiling(n/2) terms. Row sums are the Bell numbers (A000110). T(2n,n) = T(2n+1,n+1) = A000110(n) (the Bell numbers). T(n,0) = A124421(n).

			T(4,1) = 8 because we have 13|24, 1|234, 124|3, 14|2|3, 1|2|34, 13|2|4, 1|23|4 and 12|3|4.
Triangle starts:
   1;
   0,   1;
   1,   1;
   1,   3,   1;
   5,   8,   2;
   9,  26,  15,   2;
  52, 101,  45,   5;

Alois P. Heinz, Rows n = 0..150, flattened

Cf. A000110, A124418, A124419, A124421, A124422, A124423.

Q[0]:=1: for n from 1 to 13 do if n mod 2 = 1 then Q[n]:=expand(t*diff(Q[n-1],t)+x*diff(Q[n-1],s)+x*diff(Q[n-1],x)+t*Q[n-1]) else Q[n]:=expand(x*diff(Q[n-1],t)+s*diff(Q[n-1],s)+x*diff(Q[n-1],x)+s*Q[n-1]) fi od: for n from 0 to 13 do P[n]:=sort(subs({s=1,x=1},Q[n])) od: for n from 0 to 13 do seq(coeff(P[n],t,j),j=0..ceil(n/2)) od; # yields sequence in triangular form
# second Maple program:
T:= proc(n, k) local g, u; g:= floor(n/2); u:=ceil(n/2);
      add(Stirling2(i, k)*binomial(u, i)*
      add(Stirling2(g, j)*j^(u-i), j=0..g), i=k..u)
    end:
seq(seq(T(n,k), k=0..ceil(n/2)), n=0..15);  # Alois P. Heinz, Oct 23 2013

T[n_, k_] := Module[{g = Floor[n/2], u = Ceiling[n/2]},
    Sum[StirlingS2[i, k]*Binomial[u, i]*
    Sum[StirlingS2[g, j]*If[u == i, 1, j^(u - i)], {j, 0, g}], {i, k, u}]];
Table[Table[T[n, k], {k, 0, Ceiling[n/2]}], {n, 0, 15}] // Flatten (* Jean-François Alcover, May 20 2015, after Alois P. Heinz, updated Jan 01 2021 *)

The generating polynomial of row n is P[n](t)=Q[n](t,1,1), where the polynomials Q[n]=Q[n](t,s,x) are defined by Q[0]=1; Q[n]=t*dQ[n-1]/dt + x*dQ[n-1]/ds + x*dQ[n-1]/dx + t*Q[n-1] if n is odd and Q[n]=x*dQ[n-1]/dt + s*dQ[n-1]/ds + x*dQ[n-1]/dx + s*Q[n-1] if n is even.

A124422 Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n}, having exactly k blocks consisting only even entries (0<=k<=floor(n/2)).

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 5, 8, 2, 22, 25, 5, 52, 101, 45, 5, 283, 423, 156, 15, 855, 1889, 1143, 238, 15, 5451, 9726, 5002, 916, 52, 19921, 48382, 35805, 10540, 1275, 52, 144074, 292223, 187515, 49155, 5400, 203, 614866, 1609551, 1379753, 512710, 89425, 7089, 203

Offset: 0

Emeric Deutsch, Oct 31 2006

Row n has 1+floor(n/2) terms. Row sums are the Bell numbers (A000110). T(2n-1,n-1) = T(2n,n) = A000110(n) (the Bell numbers). T(n,0) = A124423(n).

			T(4,1) = 8 because we have 134|2, 13|24, 14|2|3, 1|24|3, 1|2|34, 123|4, 1|23|4 and 12|3|4.
Triangle starts:
   1;
   1;
   1,   1;
   3,   2;
   5,   8,  2;
  22,  25,  5;
  52, 101, 45, 5;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A000110, A124418, A124419, A124420, A124421, A124423.

Q[0]:=1: for n from 1 to 13 do if n mod 2 = 1 then Q[n]:=expand(t*diff(Q[n-1],t)+x*diff(Q[n-1],s)+x*diff(Q[n-1],x)+t*Q[n-1]) else Q[n]:=expand(x*diff(Q[n-1],t)+s*diff(Q[n-1],s)+x*diff(Q[n-1],x)+s*Q[n-1]) fi od: for n from 0 to 13 do P[n]:=sort(subs({t=1,x=1},Q[n])) od: for n from 0 to 13 do seq(coeff(P[n],s,j),j=0..floor(n/2)) od; # yields sequence in triangular form
# second Maple program:
T:= proc(n, k) local g, u; g:= floor(n/2); u:=ceil(n/2);
      add(Stirling2(i, k)*binomial(g, i)*
      add(Stirling2(u, j)*j^(g-i), j=0..u), i=k..g)
    end:
seq(seq(T(n,k), k=0..floor(n/2)), n=0..15); # Alois P. Heinz, Oct 23 2013

Unprotect[Power]; 0^0 = 1; T[n_, k_] := Module[{g = Floor[n/2], u = Ceiling[n/2]},  Sum[StirlingS2[i, k]*Binomial[g, i]*Sum[StirlingS2[u, j]*j^(g-i), {j, 0, u}], {i, k, g}]]; Table[Table[T[n, k], {k, 0, Floor[n/2]}], {n, 0, 15}] // Flatten (* Jean-François Alcover, May 22 2015, after Alois P. Heinz *)

The generating polynomial of row n is P[n](s)=Q[n](1,s,1), where the polynomials Q[n]=Q[n](t,s,x) are defined by Q[0]=1; Q[n]=t*dQ[n-1]/dt + x*dQ[n-1]/ds + x*dQ[n-1]/dx + t*Q[n-1] if n is odd and Q[n]=x*dQ[n-1]/dt + s*dQ[n-1]/ds + x*dQ[n-1]/dx + s*Q[n-1] if n is even.

A124423 Number of partitions of the set {1,2,...,n} having no blocks that contain only even entries.

Original entry on oeis.org

1, 1, 1, 3, 5, 22, 52, 283, 855, 5451, 19921, 144074, 614866, 4941987, 24040451, 211648665, 1152972925, 10998989896, 66200911138, 678600959525, 4465023867757, 48850849177703, 348383154017581, 4045835816532096, 31052765897026352, 381022649523561501

Offset: 0

Emeric Deutsch, Oct 31 2006

nonn

Column 0 of A124422.

			a(4) = 5 because we have 1234, 14|23, 1|234, 124|3 and 12|34.

Alois P. Heinz, Table of n, a(n) for n = 0..300

Cf. A000110, A124418, A124419, A124420, A124421, A124422.

Q[0]:=1: for n from 1 to 27 do if n mod 2 = 1 then Q[n]:=expand(t*diff(Q[n-1],t)+x*diff(Q[n-1],s)+x*diff(Q[n-1],x)+t*Q[n-1]) else Q[n]:=expand(x*diff(Q[n-1],t)+s*diff(Q[n-1],s)+x*diff(Q[n-1],x)+s*Q[n-1]) fi od: for n from 0 to 27 do Q[n]:=Q[n] od: seq(subs({t=1,s=0,x=1},Q[n]),n=0..27);
# second Maple program:
a:= n-> add(Stirling2(ceil(n/2), j)*j^floor(n/2), j=0..ceil(n/2)):
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 23 2013

a[0] = a[1] = 1; a[n_] := Sum[StirlingS2[Ceiling[n/2], j]*j^Floor[n/2], {j, 0, Ceiling[n/2]}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 22 2015, after Alois P. Heinz *)

a(n) = Q[n](1,0,1), where the polynomials Q[n]=Q[n](t,s,x) are defined by Q[0]=1; Q[n]=t*dQ[n-1]/dt + x*dQ[n-1]/ds + x*dQ[n-1]/dx + t*Q[n-1] if n is odd and Q[n]=x*dQ[n-1]/dt + s*dQ[n-1]/ds + x*dQ[n-1]/dx + s*Q[n-1] if n is even.

a(n) = Sum_{j=0..ceiling(n/2)} Stirling2(ceiling(n/2),j) * j^floor(n/2). - Alois P. Heinz, Oct 23 2013

A124425 Number of partitions of the set {1,2,...,n} having no blocks with all entries of the same parity.

Original entry on oeis.org

1, 0, 1, 1, 3, 7, 25, 79, 339, 1351, 6721, 31831, 179643, 979567, 6166105, 37852039, 262308819, 1784037031, 13471274401, 100285059751, 818288740923, 6604485845167, 57836113793305, 502235849694679, 4693153430067699, 43572170967012871, 432360767273547841

Offset: 0

Emeric Deutsch, Nov 01 2006

nonn

Column 0 of A124424.

			a(4) = 3 because we have 1234, 14|23 and 12|34.

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. A000110, A124418, A124419, A124420, A124421, A124422, A124423, A124424.

Q[0]:=1: for n from 1 to 27 do if n mod 2 = 1 then Q[n]:=expand(t*diff(Q[n-1],t)+x*diff(Q[n-1],s)+x*diff(Q[n-1],x)+t*Q[n-1]) else Q[n]:=expand(x*diff(Q[n-1],t)+s*diff(Q[n-1],s)+x*diff(Q[n-1],x)+s*Q[n-1]) fi od: seq(subs({t=0,s=0,x=1},Q[n]),n=0..27);
# second Maple program:
a:= proc(n) local g, u; g:= floor(n/2); u:= ceil(n/2);
      add(Stirling2(g, k)*Stirling2(u, k)*k!, k=0..g)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 24 2013

a[n_] := Module[{g=Floor[n/2], u=Ceiling[n/2]}, Sum[StirlingS2[g, k]*StirlingS2[u, k]*k!, {k, 0, g}]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 26 2015, after Alois P. Heinz *)

a(n) = Q[n](0,0,1), where the polynomials Q[n]=Q[n](t,s,x) are defined by Q[0]=1; Q[n]=t*dQ[n-1]/dt + x*dQ[n-1]/ds + x*dQ[n-1]/dx + t*Q[n-1] if n is odd and Q[n]=x*dQ[n-1]/dt + s*dQ[n-1]/ds + x*dQ[n-1]/dx + s*Q[n-1] if n is even.

a(n) = Sum_{k=0..floor(n/2)} Stirling2(floor(n/2),k)*Stirling2(ceiling(n/2),k)*k!. - Alois P. Heinz, Oct 24 2013

A124424 Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n}, having exactly k blocks consisting of entries of the same parity (0<=k<=n).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 2, 1, 1, 3, 4, 5, 2, 1, 7, 14, 16, 10, 4, 1, 25, 48, 61, 42, 20, 6, 1, 79, 194, 250, 200, 106, 38, 9, 1, 339, 820, 1145, 958, 569, 230, 66, 12, 1, 1351, 3794, 5554, 5096, 3251, 1486, 486, 112, 16, 1, 6721, 18960, 29101, 28010, 19110, 9470, 3477, 930, 175, 20, 1

Offset: 0

Emeric Deutsch, Nov 01 2006

Row sums are the Bell numbers (A000110). T(n,0)=A124425(n).

			T(4,2) = 5 because we have 13|24, 14|2|3, 1|2|34, 1|23|4 and 12|3|4.
Triangle starts:
  1;
  0,  1;
  1,  0,  1;
  1,  2,  1,  1;
  3,  4,  5,  2, 1;
  7, 14, 16, 10, 4, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A000110, A124418, A124419, A124420, A124421, A124422, A124423, A124425, A363434.

T(2n,n) gives A363435.

Q[0]:=1: for n from 1 to 11 do if n mod 2 = 1 then Q[n]:=expand(t*diff(Q[n-1],t)+x*diff(Q[n-1],s)+x*diff(Q[n-1],x)+t*Q[n-1]) else Q[n]:=expand(x*diff(Q[n-1],t)+s*diff(Q[n-1],s)+x*diff(Q[n-1],x)+s*Q[n-1]) fi od: for n from 0 to 11 do P[n]:=sort(subs({s=t,x=1},Q[n])) od: for n from 0 to 11 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form
# second Maple program:
b:= proc(g, u) option remember;
      add(Stirling2(g, k)*Stirling2(u, k)*k!, k=0..min(g, u))
    end:
T:= proc(n, k) local g, u; g:= floor(n/2); u:= ceil(n/2);
      add(add(add(binomial(g, i)*Stirling2(i, h)*binomial(u, j)*
      Stirling2(j, k-h)*b(g-i, u-j), j=k-h..u), i=h..g), h=0..k)
    end:
seq(seq(T(n,k), k=0..n), n=0..12);  # Alois P. Heinz, Oct 24 2013

b[g_, u_] := b[g, u] = Sum[StirlingS2[g, k]*StirlingS2[u, k]*k!, {k, 0, Min[g, u]}] ; T[n_, k_] := Module[{g, u}, g = Floor[n/2]; u = Ceiling[n/2]; Sum[ Sum[ Sum[ Binomial[g, i]*StirlingS2[i, h]*Binomial[u, j]*StirlingS2[j, k-h]*b[g-i, u-j], {j, k-h, u}], {i, h, g}], {h, 0, k}]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 19 2015, after Alois P. Heinz *)

The generating polynomial of row n is P[n](t)=Q[n](t,t,1), where the polynomials Q[n]=Q[n](t,s,x) are defined by Q[0]=1; Q[n]=t*dQ[n-1]/dt + x*dQ[n-1]/ds + x*dQ[n-1]/dx + t*Q[n-1] if n is odd and Q[n]=x*dQ[n-1]/dt + s*dQ[n-1]/ds + x*dQ[n-1]/dx + s*Q[n-1] if n is even.

Sum_{k=0..n} k * T(n,k) = A363434(n). - Alois P. Heinz, Jun 01 2023

A363429 Number of set partitions of [n] such that each block has at most one even element.

Original entry on oeis.org

1, 1, 2, 5, 10, 37, 77, 372, 799, 4736, 10427, 73013, 163967, 1322035, 3017562, 27499083, 63625324, 646147067, 1512354975, 16926317722, 40012800675, 489109544320, 1166271373797, 15455199988077, 37134022033885, 530149003318273, 1282405154139046, 19619325078384593

Offset: 0

Alois P. Heinz, Jun 01 2023

nonn

			a(0) = 1: () the empty partition.
a(1) = 1: 1.
a(2) = 2: 12, 1|2.
a(3) = 5: 123, 12|3, 13|2, 1|23, 1|2|3.
a(4) = 10: 123|4, 12|34, 12|3|4, 134|2, 13|2|4, 14|23, 1|23|4, 14|2|3, 1|2|34, 1|2|3|4.

Alois P. Heinz, Table of n, a(n) for n = 0..772
Wikipedia, Partition of a set

Bisection gives: A134980 (even part).

Cf. A000110, A110132 (exactly one even), A124421 (at least one even), A363430 (at most one odd).

b:= proc(n, m) option remember; `if`(n=0, 1,
      b(n-1, m+1)+m*b(n-1, m))
    end:
a:= n-> (h-> b(n-h, h))(iquo(n, 2)):
seq(a(n), n=0..30);

a(n) = Sum_{k=0..ceiling(n/2)} floor(n/2)^k * binomial(ceiling(n/2),k) * Bell(ceiling(n/2)-k).

cp's OEIS Frontend

A108459 Number of labeled partitions of (n,n) into pairs (i,j).

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A124419 Number of partitions of the set {1,2,...n} having no blocks that contain both odd and even entries.

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A124418 Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} having exactly k blocks that contain both odd and even entries (0<=k<=floor(n/2)).

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A124420 Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n}, having exactly k blocks consisting only odd entries (0<=k<=ceiling(n/2)).

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A124422 Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n}, having exactly k blocks consisting only even entries (0<=k<=floor(n/2)).

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A124423 Number of partitions of the set {1,2,...,n} having no blocks that contain only even entries.

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A124425 Number of partitions of the set {1,2,...,n} having no blocks with all entries of the same parity.

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