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%I #19 Mar 23 2020 17:29:58
%S 1,0,1,0,-1,1,0,7,-3,1,0,-61,31,-6,1,0,751,-375,85,-10,1,0,-11821,
%T 5911,-1350,185,-15,1,0,226927,-113463,26341,-3710,350,-21,1,0,
%U -5142061,2571031,-603246,87381,-8610,602,-28,1,0,134341711,-67170855,15887845,-2346330,240051,-17766
%N Matrix inverse of triangle A136590.
%C A136590 is the triangle of trinomial logarithmic coefficients.
%C Column 1 is signed A048287, which is the number of semiorders on n labeled nodes whose incomparability graph is connected.
%C The Bell transform of (-1)^n*A048287(n+1). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 18 2016
%F 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).
%e Triangle begins:
%e 1;
%e 0, 1;
%e 0, -1, 1;
%e 0, 7, -3, 1;
%e 0, -61, 31, -6, 1;
%e 0, 751, -375, 85, -10, 1;
%e 0, -11821, 5911, -1350, 185, -15, 1;
%e 0, 226927, -113463, 26341, -3710, 350, -21, 1;
%e 0, -5142061, 2571031, -603246, 87381, -8610, 602, -28, 1;
%e 0, 134341711, -67170855, 15887845, -2346330, 240051, -17766, 966, -36, 1; ...
%p # The function BellMatrix is defined in A264428.
%p BellMatrix(n -> (-1)^n*A048287(n+1), 9); # _Peter Luschny_, Jan 27 2016
%t BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t rows = 11;
%t M = BellMatrix[Sum[(-1)^(k+1) k! StirlingS2[#+1, k] CatalanNumber[k-1], {k, 1, #+1}]&, rows];
%t Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 24 2018, after _Peter Luschny_ *)
%o (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)))))}
%o for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
%o (PARI) /* Define Stirling2: */
%o {Stirling2(n,k) = n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!,n)}
%o /* Define Catalan(m,n) = [x^n] C(x)^m: */
%o {CATALAN(m,n) = binomial(2*n+m,n) * m/(2*n+m)}
%o /* Define this triangle: */
%o {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!))))}
%o for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
%o (Sage) # uses[bell_matrix from A264428]
%o bell_matrix(lambda n: (-1)^n*A048287(n+1), 10) # _Peter Luschny_, Jan 18 2016
%Y Cf. columns: A048287, A136596, A136597; A136590 (matrix inverse); A136588, A136589.
%K sign,tabl
%O 0,8
%A _Paul D. Hanna_, Jan 10 2008
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