A124420 - OEIS

%I #20 Jan 01 2021 11:58:44

%S 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,

%T 1889,1143,238,15,2707,8295,7320,2475,335,15,19921,48382,35805,10540,

%U 1275,52,75771,240534,240082,100940,19505,1686,52,614866,1609551,1379753,512710

%N 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)).

%C 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).

%H Alois P. Heinz, <a href="/A124420/b124420.txt">Rows n = 0..150, flattened</a>

%F 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.

%e 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.

%e Triangle starts:

%e    1;

%e    0,   1;

%e    1,   1;

%e    1,   3,   1;

%e    5,   8,   2;

%e    9,  26,  15,   2;

%e   52, 101,  45,   5;

%p 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

%p # second Maple program:

%p T:= proc(n, k) local g, u; g:= floor(n/2); u:=ceil(n/2);

%p       add(Stirling2(i, k)*binomial(u, i)*

%p       add(Stirling2(g, j)*j^(u-i), j=0..g), i=k..u)

%p     end:

%p seq(seq(T(n,k), k=0..ceil(n/2)), n=0..15);  # _Alois P. Heinz_, Oct 23 2013

%t T[n_, k_] := Module[{g = Floor[n/2], u = Ceiling[n/2]},

%t     Sum[StirlingS2[i, k]*Binomial[u, i]*

%t     Sum[StirlingS2[g, j]*If[u == i, 1, j^(u - i)], {j, 0, g}], {i, k, u}]];

%t 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 *)

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

%K nonn,tabf

%O 0,7

%A _Emeric Deutsch_, Oct 31 2006