A363434
Total number of blocks containing only elements of the same parity in all partitions of [n].
Original entry on oeis.org
0, 1, 2, 7, 24, 97, 412, 1969, 9898, 54461, 313944, 1947613, 12603100, 86760255, 620559230, 4682462777, 36586620348, 299664171115, 2534306825064, 22355119509231, 203115201624030, 1917124624702475, 18598998656476220, 186822424157036439, 1925326063016510832
Offset: 0
a(3) = 7 = 0 + 1 + 2 + 1 + 3 : 123, 12|3, 13|2, 1|23, 1|2|3.
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b:= proc(n, e, o, m) option remember; `if`(n=0, e+o,
(e+m)*b(n-1, o, e, m)+b(n-1, o, e+1, m)+
`if`(o=0, 0, o*b(n-1, o-1, e, m+1)))
end:
a:= n-> b(n, 0$3):
seq(a(n), n=0..24);
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b[n_, e_, o_, m_] := b[n, e, o, m] = If[n == 0, e + o, (e + m)*b[n-1, o, e, m] + b[n - 1, o, e + 1, m] + If[o == 0, 0, o*b[n - 1, o - 1, e, m + 1]]];
a[n_] := b[n, 0, 0, 0];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Sep 10 2023, after Alois P. Heinz *)
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
a(4) = 3 because we have 1234, 14|23 and 12|34.
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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 *)
A363435
Number of partitions of [2n] having exactly n blocks with all elements of the same parity.
Original entry on oeis.org
1, 0, 5, 42, 569, 9470, 191804, 4534502, 122544881, 3721101192, 125331498349, 4634063018948, 186515332107196, 8114659545679752, 379362605925991692, 18961051425453713478, 1008752282616284996865, 56905048753221935350268, 3392250956149146382053539
Offset: 0
a(2) = 5: 13|24, 14|2|3, 1|2|34, 1|23|4, 12|3|4.
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g:= proc(n) option remember; `if`(n=0, 1, expand(x*
add(g(n-j)*binomial(n-1, j-1), j=1..n)))
end:
S:= (n, k)-> coeff(g(n), x, k):
b:= proc(g, u) option remember;
add(S(g, k)*S(u, k)*k!, k=0..min(g, u))
end:
T:= proc(n, k) option remember; local g, u; g:= floor(n/2); u:= ceil(n/2);
add(add(add(binomial(g, i)*S(i, h)*binomial(u, j)*
S(j, k-h)*b(g-i, u-j), j=k-h..u), i=h..g), h=0..k)
end:
a:= n-> T(2*n, n):
seq(a(n), n=0..18);
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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}]];
a[n_] := T[2n, n];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Oct 20 2023, after Alois P. Heinz in A124424 *)
Showing 1-3 of 3 results.
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