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 A030528 Triangle read by rows: a(n,k) = binomial(k,n-k). 45
 1, 1, 1, 0, 2, 1, 0, 1, 3, 1, 0, 0, 3, 4, 1, 0, 0, 1, 6, 5, 1, 0, 0, 0, 4, 10, 6, 1, 0, 0, 0, 1, 10, 15, 7, 1, 0, 0, 0, 0, 5, 20, 21, 8, 1, 0, 0, 0, 0, 1, 15, 35, 28, 9, 1, 0, 0, 0, 0, 0, 6, 35, 56, 36, 10, 1, 0, 0, 0, 0, 0, 1, 21, 70, 84, 45, 11, 1, 0, 0, 0, 0, 0, 0, 7, 56, 126, 120, 55, 12, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS A convolution triangle of numbers obtained from A019590. a(n,m) := s1(-1; n,m), a member of a sequence of triangles including s1(0; n,m)= A023531(n,m) (unit matrix) and s1(2; n,m)= A007318(n-1,m-1) (Pascal's triangle). The signed triangular matrix a(n,m)*(-1)^(n-m) is the inverse matrix of the triangular Catalan convolution matrix A033184(n+1,m+1), n >= m >= 0, with A033184(n,m) := 0 if n= 2, the n-th row is also the coefficients of the vertex cover polynomial of the (n-1)-path graph P_{n-1}. - Eric W. Weisstein, Apr 10 2017 With an additional initial matrix element a_(0,0) = 1 and column of zeros a_(n,0) = 0 for n > 0, these are antidiagonals read from bottom to top of the numerical coefficients of the Maurer-Cartan form matrix of the Leibniz group L^(n)(1,1) presented on p. 9 of the Olver paper), which is generated as exp[c. * M] with (c.)^n = c_n and M the Lie infinitesimal generator A218272. Cf. A011973. And A169803. - Tom Copeland, Jul 02 2018 LINKS Indranil Ghosh, Rows 1..125 of triangle, flattened W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4. D. Merlini, R. Sprugnoli and M. C. Verri, An algebra for proper generating trees, Colloquium on Mathematics and Computer Science, Versailles, September 2000. D. Merlini, R. Sprugnoli and M. C. Verri, An algebra for proper generating trees, Mathematics and Computer Science, Part of the series Trends in Mathematics pp 127-139. P. Olver, The canonical contact form. Eric Weisstein's World of Mathematics, Path Graph Eric Weisstein's World of Mathematics, Vertex Cover Polynomial FORMULA a(n, m) = 2*(2*m-n+1)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, n

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Last modified September 20 18:49 EDT 2019. Contains 327245 sequences. (Running on oeis4.)