0,8

A136590 is the triangle of trinomial logarithmic coefficients. Column 1 is signed A048287, which is the number of semiorders on n labeled nodes whose incomparability graph is connected.

T(n,k) = Sum_{i=0..n-1} (-1)^i*(k+i)!*Stirling2(n,k+i)*Catalan(k,i)/k! where Stirling2(n,k) = A008277(n,k); Catalan(k,i) = C(2i+k,i)*k/(2i+k) = coefficient of x^i in C(x)^k with C(x) = (1-sqrt(1-4x))/(2x).

Triangle begins:

1;

0, 1;

0, -1, 1;

0, 7, -3, 1;

0, -61, 31, -6, 1;

0, 751, -375, 85, -10, 1;

0, -11821, 5911, -1350, 185, -15, 1;

0, 226927, -113463, 26341, -3710, 350, -21, 1;

0, -5142061, 2571031, -603246, 87381, -8610, 602, -28, 1;

0, 134341711, -67170855, 15887845, -2346330, 240051, -17766, 966, -36, 1; ...

(PARI) {T(n, k)=if(n<k|k<0, 0, if(n==k, 1, if(k==0, 0, n!/(k-1)!* sum(i=0, n-1, (-1)^i*polcoeff(((exp(x+x*O(x^n))-1)^(k+i)), n)*binomial(2*i+k, i)/(2*i+k)))))} (PARI) /* Define Stirling2: */ {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)} /* Define Catalan(m, n) = [x^n] C(x)^m: */ {Catalan(m, n)=binomial(2*n+m, n)*m/(2*n+m)} /* Define this triangle: */ {T(n, k)=if(n<k|k<0, 0, if(n==k, 1, if(k==0, 0, sum(i=0, n-1, (-1)^i*(k+i)!*Stirling2(n, k+i)*Catalan(k, i)/k!))))}

Cf. columns: A048287, A136596, A136597; A136590 (matrix inverse); A136588, A136589.

Adjacent sequences: A136592 A136593 A136594 this_sequence A136596 A136597 A136598

Sequence in context: A098459 A019648 A083803 this_sequence A111475 A010139 A078075

sign,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Jan 10 2008