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 A096713 Triangle of nonzero coefficients of the modified Hermite polynomials. 9
 1, 1, -1, 1, -3, 1, 3, -6, 1, 15, -10, 1, -15, 45, -15, 1, -105, 105, -21, 1, 105, -420, 210, -28, 1, 945, -1260, 378, -36, 1, -945, 4725, -3150, 630, -45, 1, -10395, 17325, -6930, 990, -55, 1, 10395, -62370, 51975, -13860, 1485, -66, 1, 135135, -270270, 135135, -25740, 2145, -78, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Triangle of nonzero coefficients of matching polynomial of complete graph of order n. Row sums of absolute values produce A000085 (number of involutions). - Wouter Meeussen, Mar 12 2008 Row n has floor(n/2)+1 nonzero coefficients. - Robert Israel, Dec 23 2015 Also the nonzero terms of the Bell matrix generated by the sequence [-1,1,0,0,0, ...] read by rows (see second Sage program). For the definition of the Bell matrix see A264428. - Peter Luschny, Jan 20 2016 REFERENCES C. D. Godsil, Algebraic Combinatorics, Chapman & Hall, New York, 1993. LINKS Robert Israel, Table of n, a(n) for n = 0..10099 (rows 0 to 199, flattened) Tom Halverson, Theodore N. Jacobson, Set-partition tableaux and representations of diagram algebras, arXiv:1808.08118 [math.RT], 2018. Eric Weisstein's World of Mathematics, Hermite Polynomial Eric Weisstein's World of Mathematics, Matching Polynomial - Eric W. Weisstein, Sep 27 2008 FORMULA G.f.: HermiteH(n,x/sqrt(2))/2^(n/2). - Wouter Meeussen, Mar 12 2008 From Robert Israel, Dec 23 2015: (Start) T(2m, k) = (-1)^(m+k)*(2m)!*2^(k-m)/((m-k)!*(2k)!), k=0..m. T(2m+1, k) = (-1)^(m+k)*(2m+1)!*2^(k-m)/((m-k)!*(2k+1)!), k=0..m. (End) EXAMPLE 1, x, -1 + x^2, -3*x + x^3, 3 - 6*x^2 + x^4, 15*x - 10*x^3 + x^5, ... MAPLE A:= NULL: for n from 0 to 20 do   HH:= expand(orthopoly[H](n, x/sqrt(2))/2^(n/2));   C:= subs(0=NULL, [seq(coeff(HH, x, j), j=0..n)]);   A:= A, op(C); od: A; #  Robert Israel, Dec 23 2015 # Alternatively: A096713 := (n, k) -> `if`(2*kn, 0, (-1)^(n\2-k)*n!/(n\2-k)!/(n%2+2*k)!/2^(n\2-k)) /* Michael Somos, Jun 04 2005 */ (Sage) from sage.functions.hypergeometric import closed_form def A096713_row(n):     R. = ZZ[]     h = hypergeometric([-n/2, (1-n)/2], [], -2*z)     T = R(closed_form(h)).coefficients()     return T[::-1] for n in range(13): A096713_row(n) # Peter Luschny, Aug 21 2014 (Sage) # Alternatively: # The function bell_transform is defined in A264428. def bell_zero_filter(generator, dim):     G = [generator(k) for k in srange(dim)]     row = lambda n: bell_transform(n, G)     F = [filter(lambda r: r != 0, R) for R in [row(n) for n in srange(dim)]]     return [i for f in F for i in f] print bell_zero_filter(lambda n: [1, -1][n] if n < 2 else 0, 14) # Peter Luschny, Jan 20 2016 (Python) from sympy import hermite, Poly, sqrt def a(n): return Poly(hermite(n, x/sqrt(2))/2**(n/2), x).coeffs()[::-1] for n in xrange(21): print a(n) # Indranil Ghosh, May 26 2017 CROSSREFS Cf. A000085. Sequence in context: A055885 A181425 A174505 * A107726 A114159 A236560 Adjacent sequences:  A096710 A096711 A096712 * A096714 A096715 A096716 KEYWORD sign,tabf AUTHOR Eric W. Weisstein, Jul 04 2004 STATUS approved

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Last modified January 16 06:59 EST 2019. Contains 319188 sequences. (Running on oeis4.)