OFFSET
1,3
COMMENTS
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.
General relation is that Poisson's kernel is the real part of this type of function (page 31 Hoffman reference).
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
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
REFERENCES
Kenneth Hoffman, Banach Spaces of Analytic Functions, Dover, New York, 1962, page 30.
Thomas McCullough, Keith Phillips, Foundations of Analysis in the Complex Plane, Holt, Reinhart and Winston, New York, 1973, 215.
S. Roman, The Umbral Calculus: Dover Publications, New York (2005).
LINKS
H. Bateman, The Polynomial of Mittag-Leffler. PNAS, 26 (8), 1940, 491-496.
Eric Weisstein's World of Mathematics, Mittag-Leffler Polynomial
FORMULA
From Peter Bala, Dec 04 2011: (Start)
T(n,k) = (-1)^k*(n-1)!*Sum{i=k..n} (-2)^i*binomial(n,i)/(i-1)!*|Stirling1(i,k)|.
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! + ....
M(n,t) = (n-1)!*Sum {k = 1..n} k*2^k*binomial(n,k)*binomial(t,k), for n>=1.
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.
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.
(End)
The row polynomials satisfy M(n, t+1) - M(n, t-1) = 2*n*M(n, t)/t. - Peter Bala, Nov 16 2016
EXAMPLE
{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, 1440, 0, 6272, 0, 2240, 0, 128},
{0, 0, 33792, 0, 39424, 0, 7168, 0, 256},
{0, 80640, 0, 418816, 0, 204288, 0, 21504, 0, 512},
{0, 0, 2594304, 0, 3676160,0, 924672, 0, 61440, 0, 1024}
MAPLE
A137513_row := proc(n) `if`(n=0, 1, 2*x*hypergeom([1-n, 1-x], [2], 2));
PolynomialTools[CoefficientList](expand(n!*simplify(%, hypergeom)), x) end:
seq(A137513_row(n), n=0..10): ListTools[FlattenOnce]([%]); # Peter Luschny, Jan 28 2016
# Alternatively, using the function BellMatrix defined in A264428:
BellMatrix(n -> `if`(n::odd, 0, 2*n!), 9); # Peter Luschny, Jan 28 2016
MATHEMATICA
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]
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: *)
MLP[0] = 1; MLP[n_] := 2x*n!*Hypergeometric2F1[1-n, 1-x, 2, 2]; Table[ CoefficientList[MLP[n], x], {n, 0, 9}] // Flatten (* or: *)
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 *)
PROG
(Sage)
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()
def A137513_row(n): return MLP(n).list()
for n in (0..9): A137513_row(n) # Peter Luschny, Jan 28 2016
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Apr 23 2008
EXTENSIONS
Edited and new name by Peter Luschny, Jan 28 2016
STATUS
approved