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 A060338 Triangle T(n,k) of coefficients of Meixner polynomials of degree n, k=0..n. 7
 1, 1, 0, 1, 0, 1, 1, 0, 5, 0, 1, 0, 14, 0, 9, 1, 0, 30, 0, 89, 0, 1, 0, 55, 0, 439, 0, 225, 1, 0, 91, 0, 1519, 0, 3429, 0, 1, 0, 140, 0, 4214, 0, 24940, 0, 11025, 1, 0, 204, 0, 10038, 0, 122156, 0, 230481, 0, 1, 0, 285, 0, 21378, 0, 463490, 0, 2250621, 0, 893025 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 COMMENTS The Meixner polynomials M_n(x) satisfy the recurrence: M_(k+1)=x*M_k-k^2*M_(k-1), M_(-1)=0, M_0=1. See A060524 for an application to combinatorics. - N. J. A. Sloane, May 30 2013 The Meixner polynomials M_n(x) satisfy: M_n(x)=n!*sum(m=0..n/2, binomial(2*m,m)*sum(j=m..n/2, (-1)^(j)*x^(n-2*j)*sum(i=0..2*j-2*m, (2^(i-2*m)*stirling1(i+n+(-2)*j,n-2*j)*binomial(n-2*m-1,2*j-2*m-i))/(i+n+(-2)*j)!))). [Vladimir Kruchinin, Sep 25 2013] REFERENCES I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983. LINKS A. Hamdi and J. Zeng. Orthogonal polynomials and operator orderings, J. Math. Phys., 51:043506, 2010; arXiv:1006.0808 [math.CO] J. Meixner, Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion, J. Lond. Math. Soc. 9 (1934), 6-13. FORMULA E.g.f.: exp(x*arctan(y))/sqrt(1+y^2). EXAMPLE [1], [1, 0], [1, 0, -1], [1, 0, -5, 0], [1, 0, -14, 0, 9], [1, 0, -30, 0, 89, 0], [1, 0, -55, 0, 439, 0, -225], [1, 0, -91, 0, 1519, 0, -3429, 0], [1, 0, -140, 0, 4214, 0, -24940, 0, 11025], [1, 0, -204, 0, 10038, 0, -122156, 0, 230481, 0], ... M_1(x)=x, M_2(x)=x^2-1, M_3(x)=x^3-5*x, M_4(x)=x^4-14*x^2+9, M_5(x)=x^5-30*x^3+89*x, M_6(x)=x^6-55*x^4+439*x^2-225,... MATHEMATICA m[0] = 1; m[1] = x; m[k_] := m[k] = x*m[k - 1] - (k - 1)^2*m[k - 2]; row[n_] := CoefficientList[m[n], x] // Reverse // Abs; Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 26 2013 *) PROG (Maxima) M(n, x):=n!*sum(binomial(2*m, m)*sum(((sum((2^(i-2*m)*stirling1(i+n-2*j, n-2*j)*binomial(n-2*m-1, 2*j-2*m-i))/(i+n-2*j)!, i, 0, 2*j-2*m))*(-1)^(j)*x^(n-2*j)), j, m, n/2), m, 0, n/2); [Vladimir Kruchinin, Sep 25 2013] CROSSREFS Cf. A028353, A060524, A000330 (third column), A214615 (row sums), A214616 (fifth column). Triangle without zeros: A094368. Unsigned version: A060524. Sequence in context: A198105 A339209 A277529 * A132795 A277031 A085198 Adjacent sequences:  A060335 A060336 A060337 * A060339 A060340 A060341 KEYWORD easy,nonn,tabl AUTHOR Vladeta Jovovic, Mar 30 2001 STATUS approved

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Last modified April 17 05:36 EDT 2021. Contains 343059 sequences. (Running on oeis4.)