A124418 -id:A124418 - cp's OEIS frontend

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

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

A124421 Number of partitions of the set {1,2,...,n} having no blocks that contain only odd entries.

Original entry on oeis.org

1, 0, 1, 1, 5, 9, 52, 130, 855, 2707, 19921, 75771, 614866, 2717570, 24040451, 120652827, 1152972925, 6460552857, 66200911138, 408845736040, 4465023867757, 30083964854141, 348383154017581, 2539795748336375, 31052765897026352, 243282175672281360

Offset: 0

Emeric Deutsch, Oct 31 2006

nonn

Column 0 of A124420.

			a(4) = 5 because we have 1234, 134|2, 14|23, 12|34 and 123|4.

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

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

Bisection gives A108459 (even part).

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=0,s=1,x=1},Q[n]),n=0..27);
# second Maple program:
a:= n-> add(Stirling2(floor(n/2), j)*j^ceil(n/2), j=0..floor(n/2)):
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 23 2013

a[0] = 1; a[n_] := Sum[StirlingS2[Floor[n/2], j]*j^Ceiling[n/2], {j, 0, Floor[n/2]}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 20 2015, after Alois P. Heinz *)

a(n) = Q[n](0,1,1), 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.

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

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

A124526 Triangle, read by rows, where T(n,k) = A049020([n/2],k)*A049020([(n+1)/2],k).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 9, 1, 10, 30, 6, 25, 100, 36, 1, 75, 370, 186, 10, 225, 1369, 961, 100, 1, 780, 5587, 4960, 750, 15, 2704, 22801, 25600, 5625, 225, 1, 10556, 101774, 136960, 39000, 2325, 21, 41209, 454276, 732736, 270400, 24025, 441, 1, 178031, 2199262, 4110512, 1849120, 217000, 6027, 28, 769129, 10647169, 23059204, 12645136, 1960000, 82369, 784, 1, 3630780, 55493841, 136074274, 87570056, 16787400, 944230, 13720, 36

Offset: 0

Paul D. Hanna, Nov 08 2006

Row n has 1+floor(n/2) terms.
T(n,0) = A124419(n).
A124418(n,k) = k!*T(n,k) (conjecture).
A000110(n) = Sum_{k=0..[n/2]} k!*T(n,k), where A000110 is the Bell numbers.
Inspired by triangle A124418 and the work of Emeric Deutsch.

			Triangle begins:
1;
1;
1, 1;
2, 3;
4, 9, 1;
10, 30, 6;
25, 100, 36, 1;
75, 370, 186, 10;
225, 1369, 961, 100, 1;
780, 5587, 4960, 750, 15;
2704, 22801, 25600, 5625, 225, 1;
10556, 101774, 136960, 39000, 2325, 21;
41209, 454276, 732736, 270400, 24025, 441, 1;
178031, 2199262, 4110512, 1849120, 217000, 6027, 28;
769129, 10647169, 23059204, 12645136, 1960000, 82369, 784, 1;
3630780, 55493841, 136074274, 87570056, 16787400, 944230, 13720, 36; ...

Cf. A124418, A124419, A049020; A124527, A124528, A124529.

S[n_, k_] = Sum[StirlingS2[n, i] Binomial[i, k], {i, 0, n}];
T[n_, k_] := S[Floor[n/2], k] S[Floor[(n+1)/2], k];
Table[T[n, k], {n, 0, 15}, {k, 0, Floor[n/2]}] // Flatten (* Jean-François Alcover, Nov 02 2020 *)

{T(n,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)}
for(n=0,15, for(k=0,n\2, print1(T(n,k),", "));print(""))

T(n,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).

A124529 a(n) = Sum_{k=0..n} k!*A124526(n+k,k) for n>=0.

Original entry on oeis.org

1, 2, 6, 29, 190, 1562, 15457, 179034, 2377092, 35599701, 593731310, 10914169312, 219252994039, 4779086510108, 112341582757512, 2833025331800643, 76293601822430388, 2185288262904326236, 66338823231846583471

Offset: 0

Paul D. Hanna, Nov 08 2006

nonn

Equals diagonal sums of triangle A124418: a(n) = Sum_{k=0..n} A124418(n+k,k) for n>=0 (conjecture).

Cf. A124526, A124527, A124528; A124418.

{a(n)=sum(k=0,n,k!*((n+k)\2)!*((n+k+1)\2)!*polcoeff(polcoeff(exp((1+y)*(exp(x+x*O(x^n))-1)),(n+k)\2),k) *polcoeff(polcoeff(exp((1+y)*(exp(x+x*O(x^n))-1)),(n+k+1)\2),k))}

A362495 Total number of blocks containing at least one odd element and at least one even element in all partitions of [n].

Original entry on oeis.org

0, 0, 1, 3, 13, 54, 262, 1294, 7109, 40367, 248651, 1587414, 10827740, 76494630, 571499993, 4414720825, 35798107309, 299547765240, 2616358573834, 23536296521084, 220030456297349, 2114721297588097, 21046291460160803, 214984439282684504, 2267305399918683232

Offset: 0

Alois P. Heinz, Jun 05 2023

nonn

			a(3) = 3 = 1 + 1 + 0 + 1 + 0 : 123, 12|3, 13|2, 1|23, 1|2|3.

Alois P. Heinz, Table of n, a(n) for n = 0..575
Wikipedia, Partition of a set

Cf. A005493, A124418, A124425, A138378, A363434.

b:= proc(n, x, y, m) option remember; `if`(n=0, m,
      `if`(x+m>0, b(n-1, y, x, m)*(x+m), 0)+b(n-1, y, x+1, m)+
      `if`(y>0, b(n-1, y-1, x, m+1)*y, 0))
    end:
a:= n-> b(n, 0$3):
seq(a(n), n=0..25);

a(n) = Sum_{k=0..floor(n/2)} k * A124418(n,k).

a(n) = A138378(n) - A363434(n) = A005493(n-1) - A363434(n) for n>=1.

cp's OEIS Frontend

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

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A124421 Number of partitions of the set {1,2,...,n} having no blocks that contain only odd entries.

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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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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).

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