A136595 Matrix inverse of triangle A136590.

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

Offset: 0

Author

Paul D. Hanna, Jan 10 2008

Keywords

• sign
• tabl

Comments

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.

The Bell transform of (-1)^n*A048287(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

Examples

			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; ...
		

Crossrefs

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

Programs

Maple

# The function BellMatrix is defined in A264428.
BellMatrix(n -> (-1)^n*A048287(n+1), 9); # Peter Luschny, Jan 27 2016

Mathematica

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 11;
M = BellMatrix[Sum[(-1)^(k+1) k! StirlingS2[#+1, k] CatalanNumber[k-1], {k, 1, #+1}]&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)

PARI

{T(n,k) = if(n
				

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
				

Sage

# uses[bell_matrix from A264428]
bell_matrix(lambda n: (-1)^n*A048287(n+1), 10) # Peter Luschny, Jan 18 2016

Formula

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).