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A094368 Triangle M(k,n) read by rows: coefficients of Meixner polynomials. 2
1, 1, -1, 1, -5, 1, -14, 9, 1, -30, 89, 1, -55, 439, -225, 1, -91, 1519, -3429, 1, -140, 4214, -24940, 11025, 1, -204, 10038, -122156, 230481, 1, -285, 21378, -463490, 2250621, -893025, 1, -385, 41778, -1467290, 14466221, -23941125, 1, -506 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,5
LINKS
Paul L. Butzer and Tom H. Koornwinder, Josef Meixner: His life and his orthogonal polynomials, Indagationes Mathematicae, Volume 30, Issue 1, January 2019, Pages 250-264.
D. Foata, Combinatoire des identités sur les polynomes de Meixner, Sem. Loth. de Comb. B06c (1982).
M. Micu, Continuous Hahn polynomials, J. Math. Phys. 34 (3) (1993), 1197-1205.
Eric Weisstein's World of Mathematics, Meixner Polynomial of the Second Kind
FORMULA
Recurrence: M(0, z) = 1, M(1, z) = z, M(n+1, z) = z*M(n, z) - n^2*M(n-1, z).
G.f.: exp(z*arctan(x)) / sqrt(1+x^2).
The n-th (unsigned) row polynomial R(n, x) = (-i)^n * M(n, i*x) = n!*Sum_{k = 0..n} 2^k*binomial(n, k)*binomial(x/2 - 1/2, k). - Peter Bala, Mar 10 2024
EXAMPLE
z,
z^2 - 1,
z^3 - 5*z,
z^4 - 14*z^2 + 9,
z^5 - 30*z^3 + 89*z,
z^6 - 55*z^4 + 439*z^2 - 225,
z^7 - 91*z^5 + 1519*z^3 - 3429*z,
z^8 - 140*z^6 + 4214*z^4 - 24940*z^2 + 11025,
z^9 - 204*z^7 + 10038*z^5 - 122156*z^3 + 230481*z,
CROSSREFS
Essentially the same as A060338.
Cf. A060524.
Sequence in context: A104792 A120393 A370518 * A295574 A087727 A039807
KEYWORD
sign,tabf
AUTHOR
Ralf Stephan, Jun 03 2004
STATUS
approved

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Last modified July 20 16:05 EDT 2024. Contains 374459 sequences. (Running on oeis4.)