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 A099174 Triangle read by rows: coefficients of modified Hermite polynomials. 13
 1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 3, 0, 6, 0, 1, 0, 15, 0, 10, 0, 1, 15, 0, 45, 0, 15, 0, 1, 0, 105, 0, 105, 0, 21, 0, 1, 105, 0, 420, 0, 210, 0, 28, 0, 1, 0, 945, 0, 1260, 0, 378, 0, 36, 0, 1, 945, 0, 4725, 0, 3150, 0, 630, 0, 45, 0, 1, 0, 10395, 0, 17325, 0, 6930, 0, 990, 0, 55 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS Absolute values of A066325. T(n,k) is the number of involutions of {1,2,...,n}, having k fixed points (0<=k<=n). Example: T(4,2)=6 because we have 1243,1432,1324,4231,3214 and 2134. - Emeric Deutsch, Oct 14 2006 Riordan array [exp(x^2/2),x]. - Paul Barry, Nov 06 2008 Same as triangle of Bessel numbers of second kind, B(n,k) (see Cheon et al., 2013). - N. J. A. Sloane, Sep 03 2013 The modified Hermite polynomial h(n,x) (as in the Formula section) is the numerator of the rational function given by f(n,x) = x + (n - 2)/f(n-1,x), where f(x,0) = 1. - Clark Kimberling, Oct 20 2014 Second lower diagonal T(n,n-2) equals positive triangular numbers A000217 \ {0}. - M. F. Hasler, Oct 23 2014 From James East, Aug 17 2015: (Start) T(n,k) is the number of R-classes (equivalently, L-classes) in the D-class consisting of all rank k elements of the Brauer monoid of degree n. For n=0 is even; 0 otherwise. - Emeric Deutsch, Oct 14 2006 G.f.: 1/(1-x*y-x^2/(1-x*y-2*x^2/(1-x*y-3*x^2/(1-x*y-4*x^2/(1-... (continued fraction). - Paul Barry, Apr 10 2009 E.g.f.: exp(y*x + x^2/2). - Geoffrey Critzer, May 08 2012 Recurrence: T(0,0)=1, T(0,k)=0 for k>0 and for n>=1 T(n,k) = T(n-1,k-1)+(k+1)*T(n-1,k+1). - Peter Luschny, Oct 06 2012 T(n+2,n) = A000217(n+1), n >= 0. - M. F. Hasler, Oct 23 2014 The row polynomials P(n,x) = (a. + x)^n, umbrally evaluated with (a.)^n = a_n = aerated A001147, are an Appell sequence with dP(n,x)/dx = n * P(n-1,x). The umbral compositional inverses (cf. A001147) of these polynomials are given by the same polynomials signed, A066325. - Tom Copeland, Nov 15 2014 From Tom Copeland, Dec 13 2015: (Start) The odd rows are (2x^2)^n x n! L(n,-1/(2x^2),1/2), and the even, (2x^2)^n n! L(n,-1/(2x^2),-1/2) in sequence with n= 0,1,2,... and L(n,x,a) = sum(k=0 to n) binom(n+a,k+a) (-x)^k/k!, the associated Laguerre polynomial of order a. The odd rows are related to A130757, and the even to A176230 and A176231. Other versions of this entry are A122848, A049403, A096713 and A104556, and reversed A100861, A144299, A111924. With each non-vanishing diagonal divided by its initial element A001147(n), this array becomes reversed, aerated A034839. Create four shift and stretch matrices S1,S2,S3, and S4 with all elements zero except S1(2n,n) = 1 for n >= 1, S2(n,2n) = 1 for n >= 0, S3(2n+1,n) = 1 for n >= 1, and S4(n,2n+1) = 1 for n >= 0. Then this entry's lower triangular matrix is T = Id + S1 * (A176230-Id) * S2 + S3 * (unsigned A130757-Id) * S4 with Id the identity matrix. The sandwiched matrices have infinitesimal generators with the nonvanishing subdiagonals A000384(n>0) and A014105(n>0). As an Appell sequence, the lowering and raising operators are L = D and R = x + dlog(exp(D^2/2))/dD = x + D, where D = d/dx, L h(n,x) = n h(n-1,x), and R h(n,x) = h(n+1,x), so R^n 1 = h(n,x). The fundamental moment sequence has the e.g.f. e^(t^2/2) with coefficients a(n) = aerated A001147, i.e., h(n,x) = (a. + x)^n, as noted above. The raising operator R as a matrix acting on o.g.f.s (formal power series) is the transpose of the production matrix P below, i.e., (1,x,x^2,...)(P^T)^n (1,0,0,...)^T = h(n,x). For charactrization as a Riordan array and associations to combinatorial structures, see the Barry link and the Yang and Qiao reference. For relations to projective modules, see the Sazdanovic link. (End) From the Appell formalism, e^(D^2/2) x^n = h_n(x), the n-th row polynomial listed below, and e^(-D^2/2) x^n = u_n(x), the n-th row polynomial of A066325. Then R = e^(D^2/2) * x * e^(-D^2/2) is another representation of the raising operator, implied by the umbral compositional inverse relation h_n(u.(x)) = x^n. - Tom Copeland, Oct 02 2016 h_n(x) = p_n(x-1), where p_n(x) are the polynomials of A111062, related to the telephone numbers A000085. - Tom Copeland, Jun 26 2018 EXAMPLE h(0,x) = 1 h(1,x) = x h(2,x) = x^2 + 1 h(3,x) = x^3 + 3*x h(4,x) = x^4 + 6*x^2 + 3 h(5,x) = x^5 + 10*x^3 + 15*x h(6,x) = x^6 + 15*x^4 + 45*x^2 + 15 From Paul Barry, Nov 06 2008: (Start) Triangle begins    1,    0,  1,    1,  0,  1,    0,  3,  0,  1,    3,  0,  6,  0,  1,    0, 15,  0, 10,  0,  1,   15,  0, 45,  0, 15,  0,  1 Production array starts   0, 1,   1, 0, 1,   0, 2, 0, 1,   0, 0, 3, 0, 1,   0, 0, 0, 4, 0, 1,   0, 0, 0, 0, 5, 0, 1 (End) MAPLE T:=proc(n, k) if n-k mod 2 = 0 then n!/2^((n-k)/2)/((n-k)/2)!/k! else 0 fi end: for n from 0 to 12 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form; Emeric Deutsch, Oct 14 2006 MATHEMATICA nn=10; a=y x+x^2/2!; Range[0, nn]!CoefficientList[Series[Exp[a], {x, 0, nn}], {x, y}]//Grid  (* Geoffrey Critzer, May 08 2102 *) H[0, x_] = 1; H[1, x_] := x; H[n_, x_] := H[n, x] = x*H[n-1, x]-(n-1)* H[n-2, x]; Table[CoefficientList[H[n, x], x], {n, 0, 11}] // Flatten // Abs (* Jean-François Alcover, May 23 2016 *) PROG (Sage) def A099174_triangle(dim):     M = matrix(SR, dim, dim)     for n in (0..dim-1): M[n, n] = 1     for n in (1..dim-1):         for k in (0..n-1):             M[n, k] = M[n-1, k-1]+(k+1)*M[n-1, k+1]     return M A099174_triangle(9)  # Peter Luschny, Oct 06 2012 (PARI) T(n, k)=if(k<=n && k==Mod(n, 2), n!/k!/(k=(n-k)/2)!>>k) \\ M. F. Hasler, Oct 23 2014 (Python) from sympy import Poly def H(n, x): return 1 if n==0 else x if n==1 else x*H(n - 1, x) - (n - 1)*H(n - 2, x) def a(n): return map(abs, Poly(H(n, x), x).all_coeffs()[::-1]) for n in xrange(21): print a(n) # Indranil Ghosh, May 26 2017 CROSSREFS Row sums (polynomial values at x=1) are A000085. Polynomial values: A005425 (x=2), A202834 (x=3), A202879(x=4). Cf. A000217, A001147, A059343, A060821, A066325. Cf. A000384. A014105, A034839, A049403, A096713, A100861, A104556, A122848, A130757, A176230, A176231. Cf. A137286. Cf. A001497. Cf. A000085, A111062. Sequence in context: A256037 A179898 A066325 * A137297 A178117 A095710 Adjacent sequences:  A099171 A099172 A099173 * A099175 A099176 A099177 KEYWORD nonn,tabl AUTHOR Ralf Stephan, on a suggestion of Karol A. Penson, Oct 13 2004 STATUS approved

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Last modified August 18 22:17 EDT 2018. Contains 313840 sequences. (Running on oeis4.)