OFFSET
0,8
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
FORMULA
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
EXAMPLE
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;
...
MAPLE
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
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Nov 01 2006
STATUS
approved