0,5

Row n has 1+floor(n/2) terms. Row sums are the Bell numbers (A000110). T(n,0)=A124419(n).

The generating polynomial of row n is P[n](x)=Q[n](1,1,x), 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.

Conjecture: T(n,k) = k!*A049020([n/2],k)*A049020([(n+1)/2],k) where A049020(n,k)=Sum_{i=0..n} S2(n,i)*C(i,k) and S2(n,k)=(1/k!)*Sum_{j=0..k} (-1)^(k-j)*C(k,j)*j^n (the Stirling numbers of 2nd kind). - Paul D. Hanna (pauldhanna(AT)juno.com), Nov 08 2006

T(4,1)=9 because we have 1234, 134|2, 1|234, 124|3, 14|2|3, 1|2|34, 123|4, 1|23|4 and 12|3|4.

Triangle starts:

1;

1;

1,1;

2,3;

4,9,2;

10,30,12;

25,100,72,6;

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, s=1}, Q[n])) od: for n from 0 to 13 do seq(coeff(P[n], x, j), j=0..floor(n/2)) od; # yields sequence in triangular form

(PARI) {T(n, k)=if(k<0|k>n, 0, k!*(n\2)!*((n+1)\2)!*polcoeff(polcoeff(exp((1+y)*(exp(x+x*O(x^n))-1)), n\2), k) *polcoeff(polcoeff(exp((1+y)*(exp(x+x*O(x^n))-1)), (n+1)\2), k))} - Paul D. Hanna (pauldhanna(AT)juno.com), Nov 08 2006

Cf. A000110, A124419, A124420, A124421, A124422, A124423.

Cf. A124526.

Sequence in context: A128657 A091930 A124526 this_sequence A112480 A112095 A091203

Adjacent sequences: A124415 A124416 A124417 this_sequence A124419 A124420 A124421

nonn,tabf

Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 31 2006