A136595 - OEIS

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,8

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

LINKS

Table of n, a(n) for n=0..51.

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).

EXAMPLE

Triangle begins:

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; ...

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 *)

PROG

(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)))))}

for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

(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!))))}

for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

(Sage) # uses[bell_matrix from A264428]

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

CROSSREFS

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