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