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Triangle read by rows: the coefficients of the Mittag-Leffler polynomials.
5

%I #51 Oct 21 2019 16:25:14

%S 1,0,2,0,0,4,0,4,0,8,0,0,32,0,16,0,48,0,160,0,32,0,0,736,0,640,0,64,0,

%T 1440,0,6272,0,2240,0,128,0,0,33792,0,39424,0,7168,0,256,0,80640,0,

%U 418816,0,204288,0,21504,0,512,0,0,2594304,0,3676160,0,924672,0,61440,0,1024

%N Triangle read by rows: the coefficients of the Mittag-Leffler polynomials.

%C Previous name was: Triangle read by rows: coefficients of the expansion of a polynomial related to the Poisson kernel: p(t,r) = ((1 + t)/(1 - t))^x: r*Exp(i*theta) -> t.

%C General relation is that Poisson's kernel is the real part of this type of function (page 31 Hoffman reference).

%C The row polynomials of this table are the Mittag-Leffler polynomials M(n,t), a polynomial sequence of binomial type [Roman, Chapter 4, Section 1.6]. The first few values are M(0,t) = 1, M(1,t) = 2*t, M(2,t) = 4*t^2, M(3,t) = 4*t+8*t^3. The polynomials M(n,t/2) are the (unsigned) row polynomials of A049218. - _Peter Bala_, Dec 04 2011

%C Also the Bell transform of the sequence "a(n) = 2*n! if n is even else 0". For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 28 2016

%D Kenneth Hoffman, Banach Spaces of Analytic Functions, Dover, New York, 1962, page 30.

%D Thomas McCullough, Keith Phillips, Foundations of Analysis in the Complex Plane, Holt, Reinhart and Winston, New York, 1973, 215.

%D S. Roman, The Umbral Calculus: Dover Publications, New York (2005).

%H Peter Bala, <a href="/A112007/a112007_Bala.txt">Diagonals of triangles with generating function exp(t*F(x)).</a>

%H H. Bateman, <a href="http://authors.library.caltech.edu/8694/">The Polynomial of Mittag-Leffler</a>. PNAS, 26 (8), 1940, 491-496.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Mittag-LefflerPolynomial.html">Mittag-Leffler Polynomial</a>

%F From _Peter Bala_, Dec 04 2011: (Start)

%F T(n,k) = (-1)^k*(n-1)!*Sum{i=k..n} (-2)^i*binomial(n,i)/(i-1)!*|Stirling1(i,k)|.

%F E.g.f.: Sum {n>=0} M(n,t)*x^n/n! = exp(t*log((1+x)/(1-x)) = ((1+x)/(1-x))^t = exp(2*t*atanh(x)) = 1 + (2*t)*x + (4*t^2)*x^2/2! + (4*t+8*t^3)*x^3/3! + ....

%F M(n,t) = (n-1)!*Sum {k = 1..n} k*2^k*binomial(n,k)*binomial(t,k), for n>=1.

%F Recurrence relation: M(n+1,t) = 2*t*Sum {k = 0..floor(n/2)} (n!/(n-2*k)!)* M(n-2*k,t), with M(0,t) = 1.

%F The o.g.f. for the n-th diagonal of the table is a rational function in t, given by the coefficient of x^n/n! in the expansion (with respect to x) of the compositional inverse (x-t*log((1+x)/(1-x)))^(-1) = x/(1-2*t) + 4*t/(1-2*t)^4*x^3/3! + (48*t+64*t^2)/(1-2*t)^7*x^5/5! + ...; for example, the o.g.f. for the fifth subdiagonal is (48*t+64*t^2)/(1-2*t)^7 = 48*t + 736*t^2 + 6272*t^3+ .... See the Bala link.

%F (End)

%F The row polynomials satisfy M(n, t+1) - M(n, t-1) = 2*n*M(n, t)/t. - _Peter Bala_, Nov 16 2016

%e {1},

%e {0, 2},

%e {0, 0, 4},

%e {0, 4, 0, 8},

%e {0, 0, 32, 0, 16},

%e {0, 48, 0, 160, 0, 32},

%e {0, 0, 736, 0, 640, 0, 64},

%e {0, 1440, 0, 6272, 0, 2240, 0, 128},

%e {0, 0, 33792, 0, 39424, 0, 7168, 0, 256},

%e {0, 80640, 0, 418816, 0, 204288, 0, 21504, 0, 512},

%e {0, 0, 2594304, 0, 3676160,0, 924672, 0, 61440, 0, 1024}

%p A137513_row := proc(n) `if`(n=0,1,2*x*hypergeom([1-n,1-x],[2],2));

%p PolynomialTools[CoefficientList](expand(n!*simplify(%,hypergeom)),x) end:

%p seq(A137513_row(n),n=0..10): ListTools[FlattenOnce]([%]); # _Peter Luschny_, Jan 28 2016

%p # Alternatively, using the function BellMatrix defined in A264428:

%p BellMatrix(n -> `if`(n::odd, 0, 2*n!), 9); # _Peter Luschny_, Jan 28 2016

%t Clear[p, f, g] p[t_] = ((1 + t)/(1 - t))^x; Table[ ExpandAll[n! * SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[ CoefficientList[n!*SeriesCoefficient[ FullSimplify[Series[p[t], {t, 0, 30} ]], n], x], {n, 0, 10}]; Flatten[a]

%t MLP[n_] := Sum[Binomial[n, k]*2^k*FactorialPower[n - 1, n - k]* FactorialPower[x, k] // FunctionExpand, {k, 0, n}]; Table[ CoefficientList[MLP[n], x], {n, 0, 9}] // Flatten (* or: *)

%t MLP[0] = 1; MLP[n_] := 2x*n!*Hypergeometric2F1[1-n, 1-x, 2, 2]; Table[ CoefficientList[MLP[n], x], {n, 0, 9}] // Flatten (* or: *)

%t BellMatrix[If[OddQ[#], 0, 2*#!]&, 9] (* in triangular matrix form, using _Peter Luschny_'s BellMatrix function defined in A264428 *) (* _Jean-François Alcover_, Jan 29 2016 *)

%o (Sage)

%o MLP = lambda n: sum(binomial(n, k)*2^k*falling_factorial(n-1, n-k)* falling_factorial(x, k) for k in (0..n)).expand()

%o def A137513_row(n): return MLP(n).list()

%o for n in (0..9): A137513_row(n) # _Peter Luschny_, Jan 28 2016

%Y Cf. A049218, A098558 (row sums).

%K nonn,tabl

%O 1,3

%A _Roger L. Bagula_, Apr 23 2008

%E Edited and new name by _Peter Luschny_, Jan 28 2016