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A137513 Triangle read by rows: the coefficients of the Mittag-Leffler polynomials.

%I A137513 #56 Feb 16 2025 08:33:07
%S A137513 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 A137513 1440,0,6272,0,2240,0,128,0,0,33792,0,39424,0,7168,0,256,0,80640,0,
%U A137513 418816,0,204288,0,21504,0,512,0,0,2594304,0,3676160,0,924672,0,61440,0,1024
%N A137513 Triangle read by rows: the coefficients of the Mittag-Leffler polynomials.
%C A137513 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 A137513 General relation is that Poisson's kernel is the real part of this type of function (page 31 Hoffman reference).
%C A137513 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 A137513 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 A137513 Kenneth Hoffman, Banach Spaces of Analytic Functions, Dover, New York, 1962, page 30.
%D A137513 Thomas McCullough, Keith Phillips, Foundations of Analysis in the Complex Plane, Holt, Reinhart and Winston, New York, 1973, 215.
%D A137513 S. Roman, The Umbral Calculus: Dover Publications, New York (2005).
%H A137513 Peter Bala, <a href="/A112007/a112007_Bala.txt">Diagonals of triangles with generating function exp(t*F(x)).</a>
%H A137513 H. Bateman, <a href="http://authors.library.caltech.edu/8694/">The Polynomial of Mittag-Leffler</a>. PNAS, 26 (8), 1940, 491-496.
%H A137513 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Mittag-LefflerPolynomial.html">Mittag-Leffler Polynomial</a>
%F A137513 From _Peter Bala_, Dec 04 2011: (Start)
%F A137513 T(n,k) = (-1)^k*(n-1)!*Sum_{i=k..n} (-2)^i*binomial(n,i)/(i-1)!*|Stirling1(i,k)|.
%F A137513 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 A137513 M(n,t) = (n-1)!*Sum_{k = 1..n} k*2^k*binomial(n,k)*binomial(t,k), for n>=1.
%F A137513 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 A137513 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 A137513 (End)
%F A137513 The row polynomials satisfy M(n, t+1) - M(n, t-1) = 2*n*M(n, t)/t. - _Peter Bala_, Nov 16 2016
%e A137513 {1},
%e A137513 {0, 2},
%e A137513 {0, 0, 4},
%e A137513 {0, 4, 0, 8},
%e A137513 {0, 0, 32, 0, 16},
%e A137513 {0, 48, 0, 160, 0, 32},
%e A137513 {0, 0, 736, 0, 640, 0, 64},
%e A137513 {0, 1440, 0, 6272, 0, 2240, 0, 128},
%e A137513 {0, 0, 33792, 0, 39424, 0, 7168, 0, 256},
%e A137513 {0, 80640, 0, 418816, 0, 204288, 0, 21504, 0, 512},
%e A137513 {0, 0, 2594304, 0, 3676160,0, 924672, 0, 61440, 0, 1024}
%p A137513 A137513_row := proc(n) `if`(n=0,1,2*x*hypergeom([1-n,1-x],[2],2));
%p A137513 PolynomialTools[CoefficientList](expand(n!*simplify(%,hypergeom)),x) end:
%p A137513 seq(A137513_row(n),n=0..10): ListTools[FlattenOnce]([%]); # _Peter Luschny_, Jan 28 2016
%p A137513 # Alternatively, using the function BellMatrix defined in A264428:
%p A137513 BellMatrix(n -> `if`(n::odd, 0, 2*n!), 9); # _Peter Luschny_, Jan 28 2016
%t A137513 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 A137513 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 A137513 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 A137513 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 A137513 (Sage)
%o A137513 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 A137513 def A137513_row(n): return MLP(n).list()
%o A137513 for n in (0..9): A137513_row(n) # _Peter Luschny_, Jan 28 2016
%Y A137513 Cf. A049218, A098558 (row sums).
%K A137513 nonn,tabl
%O A137513 1,3
%A A137513 _Roger L. Bagula_, Apr 23 2008
%E A137513 Edited and new name by _Peter Luschny_, Jan 28 2016