A137378 Triangle of the coefficient [x^k] of the polynomial 2^n*s_n(x) generated by exp(x*(1 - sqrt(1+t^2))/t) = sum_{n>=0} s_n(x)*t^k/k! in row n, column k.

1, 0, -1, 0, 0, 1, 0, 6, 0, -1, 0, 0, -24, 0, 1, 0, -240, 0, 60, 0, -1, 0, 0, 1800, 0, -120, 0, 1, 0, 25200, 0, -7560, 0, 210, 0, -1, 0, 0, -282240, 0, 23520, 0, -336, 0, 1, 0, -5080320, 0, 1693440, 0, -60480, 0, 504, 0, -1, 0, 0, 76204800, 0, -7257600, 0, 136080, 0, -720, 0, 1

Offset

0,8

Comments

Row sums are: 1, -1, 1, 5, -23, -181, 1681, 17849, -259055, -3446857, 69082561,..

Weisstein uses the nomenclature "Mott Polynomial" for s_n(x), although his definition differs from Mott's by signs. - R. J. Mathar, Oct 03 2013

Also the Bell transform of the sequence defined below in the Maple program. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

Links

Table of n, a(n) for n=0..65.

Ömür Kıvanç Kürkçü, A new numerical method for solving delay integral equations with variable bounds by using generalized Mott polynomials, Anadolu University Journal of Science and Technology A, Applied Sciences and Engineering (2018) Vol. 19, No. 4, 264-277.

Ömür Kıvanç Kürkçü, A numerical method with a control parameter for integro-differential delay equations with state-dependent bounds via generalized Mott polynomial, Mathematical Sciences (2019).

N. F. Mott, The polarisation of electrons by double scattering, Proc. R. Soc. Lond. A 135 (827) (1932), p 442.

Eric Weisstein's MathWorld, Mott Polynomial

Wikipedia, Mott Polynomials

Formula

p(x) = Exp[x*(1 - Sqrt[1 + t^2])/t]; weights 2^n*n!;

M(n,x) = n!/2^n *sum_{m=floor((n+1)/2)..n} ((-1)^m *(2*m-n) *binomial(n-1,m-1) *x^(2*m-n))/(m*(2*m-n)!). [Dmitry Kruchinin Mar 24 2013]

Example

1;
0, -1;
0, 0, 1;
0, 6, 0, -1;
0, 0, -24, 0, 1;
0, -240,0, 60, 0, -1;
0, 0, 1800, 0, -120, 0, 1;
0, 25200, 0, -7560, 0, 210, 0, -1;
0, 0, -282240, 0, 23520, 0, -336, 0, 1;
0, -5080320, 0, 1693440, 0, -60480, 0, 504, 0, -1;
0, 0, 76204800, 0, -7257600, 0, 136080, 0, -720, 0, 1;

Maple

# The function BellMatrix is defined in A264428.
BellMatrix(n -> `if`(n::odd, 0, (-1)^(1+n/2)*(n+1)/(n/2+1)*(n!/(n/2)!)^2), 9); # Peter Luschny, Jan 27 2016
A137378 := proc(n, k)
    local m ;
    if n =0 and k =0 then
        1;
    elif type(n+k, 'odd') then
        0;
    else
        m := (n+k)/2 ;
        (-1)^m*k*binomial(n-1, m-1)/m/k! ;
        %*n! ;
    end if;
end proc: # R. J. Mathar, Nov 17 2018

Mathematica

p[t_] = Exp[x*(1 - Sqrt[1 + t^2])/t]; Table[ ExpandAll[2^(n)*n! * SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[ CoefficientList[2^(n)*n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}]; Flatten[a]
(* Second program: *)
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[Function[n, If[OddQ[n], 0, (-1)^(1 + n/2)*(n + 1)/(n/2 + 1)*(n!/(n/2)!)^2]], rows = 12];
Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

Prog

(Maxima)
M(n):=n!*sum(((2*m-n)*(-1)^(m)*binomial(n-1, m-1)*x^(2*m-n)/((2*m-n)!*(m))), m, floor((n+1)/2), n);
for n:0 thru 7  do if n=0 then print([1]) else (LL:makelist(coeff(ratsimp(M(n)), x, k), k, 0, n), print(LL)); // Dmitry Kruchinin, Mar 24 2013

Crossrefs

Sequence in context: A240315 A339431 A256041 * A333275 A293071 A084680

Adjacent sequences: A137375 A137376 A137377 * A137379 A137380 A137381

Keyword

tabl,sign

Author

Roger L. Bagula, Apr 09 2008

Status

approved