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)).
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
Examples
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;
Links
- Alois P. Heinz, Rows n = 0..150, flattened
Programs
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Maple
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 -
Mathematica
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 *)
Formula
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.
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