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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)).

%I A124422 #21 Jun 02 2023 16:24:30
%S A124422 1,1,1,1,3,2,5,8,2,22,25,5,52,101,45,5,283,423,156,15,855,1889,1143,
%T A124422 238,15,5451,9726,5002,916,52,19921,48382,35805,10540,1275,52,144074,
%U A124422 292223,187515,49155,5400,203,614866,1609551,1379753,512710,89425,7089,203
%N 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)).
%C A124422 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).
%H A124422 Alois P. Heinz, <a href="/A124422/b124422.txt">Rows n = 0..200, flattened</a>
%F A124422 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.
%e A124422 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.
%e A124422 Triangle starts:
%e A124422    1;
%e A124422    1;
%e A124422    1,   1;
%e A124422    3,   2;
%e A124422    5,   8,  2;
%e A124422   22,  25,  5;
%e A124422   52, 101, 45, 5;
%e A124422   ...
%p A124422 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
%p A124422 # second Maple program:
%p A124422 T:= proc(n, k) local g, u; g:= floor(n/2); u:=ceil(n/2);
%p A124422       add(Stirling2(i, k)*binomial(g, i)*
%p A124422       add(Stirling2(u, j)*j^(g-i), j=0..u), i=k..g)
%p A124422     end:
%p A124422 seq(seq(T(n,k), k=0..floor(n/2)), n=0..15); # _Alois P. Heinz_, Oct 23 2013
%t A124422 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_ *)
%Y A124422 Cf. A000110, A124418, A124419, A124420, A124421, A124423.
%K A124422 nonn,tabf
%O A124422 0,5
%A A124422 _Emeric Deutsch_, Oct 31 2006